[{"data":1,"prerenderedAt":15616},["ShallowReactive",2],{"note:math\u002Fabstract_algebra\u002Fringdirectprod":3,"site-search-catalogue":15615},{"id":4,"title":5,"body":6,"description":16,"extension":15607,"meta":15608,"navigation":15610,"path":15611,"seo":15612,"stem":15613,"__hash__":15614},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fringdirectprod.md","环的直积与中国剩余定理",{"type":7,"value":8,"toc":15598},"minimark",[9,13,17,34,37,42,45,50,183,1982,1986,2152,2323,2325,2329,2332,2845,3172,3176,3179,3624,3827,3832,4768,4863,4865,4869,5056,5061,13253,13255,13259,13325,14991,14993,14996],[10,11,12],"h1",{"id":12},"环的直积直和与中国剩余定理",[14,15,16],"p",{},"核心在于讲清楚两件事：",[18,19,20,28],"ol",{},[21,22,23,27],"li",{},[24,25,26],"strong",{},"定义工具","：什么是直积和直和（把小环拼成大环）。",[21,29,30,33],{},[24,31,32],{},"核心定理","：中国剩余定理（CRT）—— 如何把一个大环的问题，拆解成几个小环的问题来解，并且证明它们是等价的。",[35,36],"hr",{},[38,39,41],"h3",{"id":40},"第一阶段弄清楚概念直积与直和","第一阶段：弄清楚概念（直积与直和）",[14,43,44],{},"在进入复杂的证明前，我们必须先看懂符号。",[46,47,49],"h4",{"id":48},"_1-定义-341直积-direct-product","1. 定义 3.4.1：直积 (Direct Product)",[14,51,52,55,56,179,182],{},[24,53,54],{},"符号","：",[57,58,61,94],"span",{"className":59},[60],"katex",[57,62,65],{"className":63},[64],"katex-mathml",[66,67,69],"math",{"xmlns":68},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[70,71,72,89],"semantics",{},[73,74,75,79],"mrow",{},[76,77,78],"mo",{},"∏",[80,81,82,86],"msub",{},[83,84,85],"mi",{},"R",[83,87,88],{},"i",[90,91,93],"annotation",{"encoding":92},"application\u002Fx-tex","\\prod 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R_3...",[57,312,314],{"className":313,"ariaHidden":98},[97],[57,315,317,321,362,366,369,409,412,415,455],{"className":316},[102],[57,318],{"className":319,"style":320},[106],"height:0.8778em;vertical-align:-0.1944em;",[57,322,324,327],{"className":323},[123],[57,325,85],{"className":326,"style":128},[123,127],[57,328,330],{"className":329},[132],[57,331,333,354],{"className":332},[136,137],[57,334,336,351],{"className":335},[141],[57,337,340],{"className":338,"style":339},[145],"height:0.3011em;",[57,341,342,345],{"style":149},[57,343],{"className":344,"style":154},[153],[57,346,348],{"className":347},[158,159,160,161],[57,349,280],{"className":350},[123,161],[57,352,169],{"className":353},[168],[57,355,357],{"className":356},[141],[57,358,360],{"className":359,"style":176},[145],[57,361],{},[57,363,283],{"className":364},[365],"mpunct",[57,367],{"className":368,"style":119},[118],[57,370,372,375],{"className":371},[123],[57,373,85],{"className":374,"style":128},[123,127],[57,376,378],{"className":377},[132],[57,379,381,401],{"className":380},[136,137],[57,382,384,398],{"className":383},[141],[57,385,387],{"className":386,"style":339},[145],[57,388,389,392],{"style":149},[57,390],{"className":391,"style":154},[153],[57,393,395],{"className":394},[158,159,160,161],[57,396,290],{"className":397},[123,161],[57,399,169],{"className":400},[168],[57,402,404],{"className":403},[141],[57,405,407],{"className":406,"style":176},[145],[57,408],{},[57,410,283],{"className":411},[365],[57,413],{"className":414,"style":119},[118],[57,416,418,421],{"className":417},[123],[57,419,85],{"className":420,"style":128},[123,127],[57,422,424],{"className":423},[132],[57,425,427,447],{"className":426},[136,137],[57,428,430,444],{"className":429},[141],[57,431,433],{"className":432,"style":339},[145],[57,434,435,438],{"style":149},[57,436],{"className":437,"style":154},[153],[57,439,441],{"className":440},[158,159,160,161],[57,442,299],{"className":443},[123,161],[57,445,169],{"className":446},[168],[57,448,450],{"className":449},[141],[57,451,453],{"className":452,"style":176},[145],[57,454],{},[57,456,458],{"className":457},[123],"...","）。",[21,461,462,463,680,681,752,753,823,824,752,894,964],{},"直积里的一个元素长这样：",[57,464,466,517],{"className":465},[60],[57,467,469],{"className":468},[64],[66,470,471],{"xmlns":68},[70,472,473,514],{},[73,474,475,479,486,488,494,496,502,504,507,511],{},[76,476,478],{"stretchy":477},"false","(",[80,480,481,484],{},[83,482,483],{},"a",[278,485,280],{},[76,487,283],{"separator":98},[80,489,490,492],{},[83,491,483],{},[278,493,290],{},[76,495,283],{"separator":98},[80,497,498,500],{},[83,499,483],{},[278,501,299],{},[76,503,283],{"separator":98},[76,505,506],{},"…",[508,509,510],"mtext",{}," ",[76,512,513],{"stretchy":477},")",[90,515,516],{"encoding":92},"(a_1, a_2, a_3, \\dots)",[57,518,520],{"className":519,"ariaHidden":98},[97],[57,521,523,526,530,571,574,577,617,620,623,663,666,669,673,676],{"className":522},[102],[57,524],{"className":525,"style":107},[106],[57,527,478],{"className":528},[529],"mopen",[57,531,533,536],{"className":532},[123],[57,534,483],{"className":535},[123,127],[57,537,539],{"className":538},[132],[57,540,542,563],{"className":541},[136,137],[57,543,545,560],{"className":544},[141],[57,546,548],{"className":547,"style":339},[145],[57,549,551,554],{"style":550},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[57,552],{"className":553,"style":154},[153],[57,555,557],{"className":556},[158,159,160,161],[57,558,280],{"className":559},[123,161],[57,561,169],{"className":562},[168],[57,564,566],{"className":565},[141],[57,567,569],{"className":568,"style":176},[145],[57,570],{},[57,572,283],{"className":573},[365],[57,575],{"className":576,"style":119},[118],[57,578,580,583],{"className":579},[123],[57,581,483],{"className":582},[123,127],[57,584,586],{"className":585},[132],[57,587,589,609],{"className":588},[136,137],[57,590,592,606],{"className":591},[141],[57,593,595],{"className":594,"style":339},[145],[57,596,597,600],{"style":550},[57,598],{"className":599,"style":154},[153],[57,601,603],{"className":602},[158,159,160,161],[57,604,290],{"className":605},[123,161],[57,607,169],{"className":608},[168],[57,610,612],{"className":611},[141],[57,613,615],{"className":614,"style":176},[145],[57,616],{},[57,618,283],{"className":619},[365],[57,621],{"className":622,"style":119},[118],[57,624,626,629],{"className":625},[123],[57,627,483],{"className":628},[123,127],[57,630,632],{"className":631},[132],[57,633,635,655],{"className":634},[136,137],[57,636,638,652],{"className":637},[141],[57,639,641],{"className":640,"style":339},[145],[57,642,643,646],{"style":550},[57,644],{"className":645,"style":154},[153],[57,647,649],{"className":648},[158,159,160,161],[57,650,299],{"className":651},[123,161],[57,653,169],{"className":654},[168],[57,656,658],{"className":657},[141],[57,659,661],{"className":660,"style":176},[145],[57,662],{},[57,664,283],{"className":665},[365],[57,667],{"className":668,"style":119},[118],[57,670,506],{"className":671},[672],"minner",[57,674],{"className":675,"style":119},[118],[57,677,513],{"className":678},[679],"mclose","，其中 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",[57,754,756,774],{"className":755},[60],[57,757,759],{"className":758},[64],[66,760,761],{"xmlns":68},[70,762,763,771],{},[73,764,765],{},[80,766,767,769],{},[83,768,85],{},[278,770,280],{},[90,772,773],{"encoding":92},"R_1",[57,775,777],{"className":776,"ariaHidden":98},[97],[57,778,780,783],{"className":779},[102],[57,781],{"className":782,"style":219},[106],[57,784,786,789],{"className":785},[123],[57,787,85],{"className":788,"style":128},[123,127],[57,790,792],{"className":791},[132],[57,793,795,815],{"className":794},[136,137],[57,796,798,812],{"className":797},[141],[57,799,801],{"className":800,"style":339},[145],[57,802,803,806],{"style":149},[57,804],{"className":805,"style":154},[153],[57,807,809],{"className":808},[158,159,160,161],[57,810,280],{"className":811},[123,161],[57,813,169],{"className":814},[168],[57,816,818],{"className":817},[141],[57,819,821],{"className":820,"style":176},[145],[57,822],{},"， ",[57,825,827,845],{"className":826},[60],[57,828,830],{"className":829},[64],[66,831,832],{"xmlns":68},[70,833,834,842],{},[73,835,836],{},[80,837,838,840],{},[83,839,483],{},[278,841,290],{},[90,843,844],{"encoding":92},"a_2",[57,846,848],{"className":847,"ariaHidden":98},[97],[57,849,851,854],{"className":850},[102],[57,852],{"className":853,"style":711},[106],[57,855,857,860],{"className":856},[123],[57,858,483],{"className":859},[123,127],[57,861,863],{"className":862},[132],[57,864,866,886],{"className":865},[136,137],[57,867,869,883],{"className":868},[141],[57,870,872],{"className":871,"style":339},[145],[57,873,874,877],{"style":550},[57,875],{"className":876,"style":154},[153],[57,878,880],{"className":879},[158,159,160,161],[57,881,290],{"className":882},[123,161],[57,884,169],{"className":885},[168],[57,887,889],{"className":888},[141],[57,890,892],{"className":891,"style":176},[145],[57,893],{},[57,895,897,915],{"className":896},[60],[57,898,900],{"className":899},[64],[66,901,902],{"xmlns":68},[70,903,904,912],{},[73,905,906],{},[80,907,908,910],{},[83,909,85],{},[278,911,290],{},[90,913,914],{"encoding":92},"R_2",[57,916,918],{"className":917,"ariaHidden":98},[97],[57,919,921,924],{"className":920},[102],[57,922],{"className":923,"style":219},[106],[57,925,927,930],{"className":926},[123],[57,928,85],{"className":929,"style":128},[123,127],[57,931,933],{"className":932},[132],[57,934,936,956],{"className":935},[136,137],[57,937,939,953],{"className":938},[141],[57,940,942],{"className":941,"style":339},[145],[57,943,944,947],{"style":149},[57,945],{"className":946,"style":154},[153],[57,948,950],{"className":949},[158,159,160,161],[57,951,290],{"className":952},[123,161],[57,954,169],{"className":955},[168],[57,957,959],{"className":958},[141],[57,960,962],{"className":961,"style":176},[145],[57,963],{},"，以此类推。",[21,966,967,970,971],{},[24,968,969],{},"运算规则","：就是“各算各的”。\n",[184,972,973,1498],{},[21,974,975,976],{},"加法：",[57,977,979,1068],{"className":978},[60],[57,980,982],{"className":981},[64],[66,983,984],{"xmlns":68},[70,985,986,1065],{},[73,987,988,990,996,998,1004,1006,1009,1011,1018,1020,1026,1028,1031,1033,1039,1041,1047,1049,1055,1057,1063],{},[76,989,478],{"stretchy":477},[80,991,992,994],{},[83,993,483],{},[278,995,280],{},[76,997,283],{"separator":98},[80,999,1000,1002],{},[83,1001,483],{},[278,1003,290],{},[76,1005,513],{"stretchy":477},[76,1007,1008],{},"+",[76,1010,478],{"stretchy":477},[80,1012,1013,1016],{},[83,1014,1015],{},"b",[278,1017,280],{},[76,1019,283],{"separator":98},[80,1021,1022,1024],{},[83,1023,1015],{},[278,1025,290],{},[76,1027,513],{"stretchy":477},[76,1029,1030],{},"=",[76,1032,478],{"stretchy":477},[80,1034,1035,1037],{},[83,1036,483],{},[278,1038,280],{},[76,1040,1008],{},[80,1042,1043,1045],{},[83,1044,1015],{},[278,1046,280],{},[76,1048,283],{"separator":98},[80,1050,1051,1053],{},[83,1052,483],{},[278,1054,290],{},[76,1056,1008],{},[80,1058,1059,1061],{},[83,1060,1015],{},[278,1062,290],{},[76,1064,513],{"stretchy":477},[90,1066,1067],{"encoding":92},"(a_1, a_2) + (b_1, b_2) = (a_1+b_1, a_2+b_2)",[57,1069,1071,1180,1289,1347,1449],{"className":1070,"ariaHidden":98},[97],[57,1072,1074,1077,1080,1120,1123,1126,1166,1169,1173,1177],{"className":1073},[102],[57,1075],{"className":1076,"style":107},[106],[57,1078,478],{"className":1079},[529],[57,1081,1083,1086],{"className":1082},[123],[57,1084,483],{"className":1085},[123,127],[57,1087,1089],{"className":1088},[132],[57,1090,1092,1112],{"className":1091},[136,137],[57,1093,1095,1109],{"className":1094},[141],[57,1096,1098],{"className":1097,"style":339},[145],[57,1099,1100,1103],{"style":550},[57,1101],{"className":1102,"style":154},[153],[57,1104,1106],{"className":1105},[158,159,160,161],[57,1107,280],{"className":1108},[123,161],[57,1110,169],{"className":1111},[168],[57,1113,1115],{"className":1114},[141],[57,1116,1118],{"className":1117,"style":176},[145],[57,1119],{},[57,1121,283],{"className":1122},[365],[57,1124],{"className":1125,"style":119},[118],[57,1127,1129,1132],{"className":1128},[123],[57,1130,483],{"className":1131},[123,127],[57,1133,1135],{"className":1134},[132],[57,1136,1138,1158],{"className":1137},[136,137],[57,1139,1141,1155],{"className":1140},[141],[57,1142,1144],{"className":1143,"style":339},[145],[57,1145,1146,1149],{"style":550},[57,1147],{"className":1148,"style":154},[153],[57,1150,1152],{"className":1151},[158,159,160,161],[57,1153,290],{"className":1154},[123,161],[57,1156,169],{"className":1157},[168],[57,1159,1161],{"className":1160},[141],[57,1162,1164],{"className":1163,"style":176},[145],[57,1165],{},[57,1167,513],{"className":1168},[679],[57,1170],{"className":1171,"style":1172},[118],"margin-right:0.2222em;",[57,1174,1008],{"className":1175},[1176],"mbin",[57,1178],{"className":1179,"style":1172},[118],[57,1181,1183,1186,1189,1229,1232,1235,1275,1278,1282,1286],{"className":1182},[102],[57,1184],{"className":1185,"style":107},[106],[57,1187,478],{"className":1188},[529],[57,1190,1192,1195],{"className":1191},[123],[57,1193,1015],{"className":1194},[123,127],[57,1196,1198],{"className":1197},[132],[57,1199,1201,1221],{"className":1200},[136,137],[57,1202,1204,1218],{"className":1203},[141],[57,1205,1207],{"className":1206,"style":339},[145],[57,1208,1209,1212],{"style":550},[57,1210],{"className":1211,"style":154},[153],[57,1213,1215],{"className":1214},[158,159,160,161],[57,1216,280],{"className":1217},[123,161],[57,1219,169],{"className":1220},[168],[57,1222,1224],{"className":1223},[141],[57,1225,1227],{"className":1226,"style":176},[145],[57,1228],{},[57,1230,283],{"className":1231},[365],[57,1233],{"className":1234,"style":119},[118],[57,1236,1238,1241],{"className":1237},[123],[57,1239,1015],{"className":1240},[123,127],[57,1242,1244],{"className":1243},[132],[57,1245,1247,1267],{"className":1246},[136,137],[57,1248,1250,1264],{"className":1249},[141],[57,1251,1253],{"className":1252,"style":339},[145],[57,1254,1255,1258],{"style":550},[57,1256],{"className":1257,"style":154},[153],[57,1259,1261],{"className":1260},[158,159,160,161],[57,1262,290],{"className":1263},[123,161],[57,1265,169],{"className":1266},[168],[57,1268,1270],{"className":1269},[141],[57,1271,1273],{"className":1272,"style":176},[145],[57,1274],{},[57,1276,513],{"className":1277},[679],[57,1279],{"className":1280,"style":1281},[118],"margin-right:0.2778em;",[57,1283,1030],{"className":1284},[1285],"mrel",[57,1287],{"className":1288,"style":1281},[118],[57,1290,1292,1295,1298,1338,1341,1344],{"className":1291},[102],[57,1293],{"className":1294,"style":107},[106],[57,1296,478],{"className":1297},[529],[57,1299,1301,1304],{"className":1300},[123],[57,1302,483],{"className":1303},[123,127],[57,1305,1307],{"className":1306},[132],[57,1308,1310,1330],{"className":1309},[136,137],[57,1311,1313,1327],{"className":1312},[141],[57,1314,1316],{"className":1315,"style":339},[145],[57,1317,1318,1321],{"style":550},[57,1319],{"className":1320,"style":154},[153],[57,1322,1324],{"className":1323},[158,159,160,161],[57,1325,280],{"className":1326},[123,161],[57,1328,169],{"className":1329},[168],[57,1331,1333],{"className":1332},[141],[57,1334,1336],{"className":1335,"style":176},[145],[57,1337],{},[57,1339],{"className":1340,"style":1172},[118],[57,1342,1008],{"className":1343},[1176],[57,1345],{"className":1346,"style":1172},[118],[57,1348,1350,1354,1394,1397,1400,1440,1443,1446],{"className":1349},[102],[57,1351],{"className":1352,"style":1353},[106],"height:0.8889em;vertical-align:-0.1944em;",[57,1355,1357,1360],{"className":1356},[123],[57,1358,1015],{"className":1359},[123,127],[57,1361,1363],{"className":1362},[132],[57,1364,1366,1386],{"className":1365},[136,137],[57,1367,1369,1383],{"className":1368},[141],[57,1370,1372],{"className":1371,"style":339},[145],[57,1373,1374,1377],{"style":550},[57,1375],{"className":1376,"style":154},[153],[57,1378,1380],{"className":1379},[158,159,160,161],[57,1381,280],{"className":1382},[123,161],[57,1384,169],{"className":1385},[168],[57,1387,1389],{"className":1388},[141],[57,1390,1392],{"className":1391,"style":176},[145],[57,1393],{},[57,1395,283],{"className":1396},[365],[57,1398],{"className":1399,"style":119},[118],[57,1401,1403,1406],{"className":1402},[123],[57,1404,483],{"className":1405},[123,127],[57,1407,1409],{"className":1408},[132],[57,1410,1412,1432],{"className":1411},[136,137],[57,1413,1415,1429],{"className":1414},[141],[57,1416,1418],{"className":1417,"style":339},[145],[57,1419,1420,1423],{"style":550},[57,1421],{"className":1422,"style":154},[153],[57,1424,1426],{"className":1425},[158,159,160,161],[57,1427,290],{"className":1428},[123,161],[57,1430,169],{"className":1431},[168],[57,1433,1435],{"className":1434},[141],[57,1436,1438],{"className":1437,"style":176},[145],[57,1439],{},[57,1441],{"className":1442,"style":1172},[118],[57,1444,1008],{"className":1445},[1176],[57,1447],{"className":1448,"style":1172},[118],[57,1450,1452,1455,1495],{"className":1451},[102],[57,1453],{"className":1454,"style":107},[106],[57,1456,1458,1461],{"className":1457},[123],[57,1459,1015],{"className":1460},[123,127],[57,1462,1464],{"className":1463},[132],[57,1465,1467,1487],{"className":1466},[136,137],[57,1468,1470,1484],{"className":1469},[141],[57,1471,1473],{"className":1472,"style":339},[145],[57,1474,1475,1478],{"style":550},[57,1476],{"className":1477,"style":154},[153],[57,1479,1481],{"className":1480},[158,159,160,161],[57,1482,290],{"className":1483},[123,161],[57,1485,169],{"className":1486},[168],[57,1488,1490],{"className":1489},[141],[57,1491,1493],{"className":1492,"style":176},[145],[57,1494],{},[57,1496,513],{"className":1497},[679],[21,1499,1500,1501],{},"乘法：",[57,1502,1504,1587],{"className":1503},[60],[57,1505,1507],{"className":1506},[64],[66,1508,1509],{"xmlns":68},[70,1510,1511,1584],{},[73,1512,1513,1515,1521,1523,1529,1531,1534,1536,1542,1544,1550,1552,1554,1556,1562,1568,1570,1576,1582],{},[76,1514,478],{"stretchy":477},[80,1516,1517,1519],{},[83,1518,483],{},[278,1520,280],{},[76,1522,283],{"separator":98},[80,1524,1525,1527],{},[83,1526,483],{},[278,1528,290],{},[76,1530,513],{"stretchy":477},[76,1532,1533],{},"⋅",[76,1535,478],{"stretchy":477},[80,1537,1538,1540],{},[83,1539,1015],{},[278,1541,280],{},[76,1543,283],{"separator":98},[80,1545,1546,1548],{},[83,1547,1015],{},[278,1549,290],{},[76,1551,513],{"stretchy":477},[76,1553,1030],{},[76,1555,478],{"stretchy":477},[80,1557,1558,1560],{},[83,1559,483],{},[278,1561,280],{},[80,1563,1564,1566],{},[83,1565,1015],{},[278,1567,280],{},[76,1569,283],{"separator":98},[80,1571,1572,1574],{},[83,1573,483],{},[278,1575,290],{},[80,1577,1578,1580],{},[83,1579,1015],{},[278,1581,290],{},[76,1583,513],{"stretchy":477},[90,1585,1586],{"encoding":92},"(a_1, a_2) \\cdot (b_1, b_2) = (a_1 b_1, a_2 b_2)",[57,1588,1590,1697,1804],{"className":1589,"ariaHidden":98},[97],[57,1591,1593,1596,1599,1639,1642,1645,1685,1688,1691,1694],{"className":1592},[102],[57,1594],{"className":1595,"style":107},[106],[57,1597,478],{"className":1598},[529],[57,1600,1602,1605],{"className":1601},[123],[57,1603,483],{"className":1604},[123,127],[57,1606,1608],{"className":1607},[132],[57,1609,1611,1631],{"className":1610},[136,137],[57,1612,1614,1628],{"className":1613},[141],[57,1615,1617],{"className":1616,"style":339},[145],[57,1618,1619,1622],{"style":550},[57,1620],{"className":1621,"style":154},[153],[57,1623,1625],{"className":1624},[158,159,160,161],[57,1626,280],{"className":1627},[123,161],[57,1629,169],{"className":1630},[168],[57,1632,1634],{"className":1633},[141],[57,1635,1637],{"className":1636,"style":176},[145],[57,1638],{},[57,1640,283],{"className":1641},[365],[57,1643],{"className":1644,"style":119},[118],[57,1646,1648,1651],{"className":1647},[123],[57,1649,483],{"className":1650},[123,127],[57,1652,1654],{"className":1653},[132],[57,1655,1657,1677],{"className":1656},[136,137],[57,1658,1660,1674],{"className":1659},[141],[57,1661,1663],{"className":1662,"style":339},[145],[57,1664,1665,1668],{"style":550},[57,1666],{"className":1667,"style":154},[153],[57,1669,1671],{"className":1670},[158,159,160,161],[57,1672,290],{"className":1673},[123,161],[57,1675,169],{"className":1676},[168],[57,1678,1680],{"className":1679},[141],[57,1681,1683],{"className":1682,"style":176},[145],[57,1684],{},[57,1686,513],{"className":1687},[679],[57,1689],{"className":1690,"style":1172},[118],[57,1692,1533],{"className":1693},[1176],[57,1695],{"className":1696,"style":1172},[118],[57,1698,1700,1703,1706,1746,1749,1752,1792,1795,1798,1801],{"className":1699},[102],[57,1701],{"className":1702,"style":107},[106],[57,1704,478],{"className":1705},[529],[57,1707,1709,1712],{"className":1708},[123],[57,1710,1015],{"className":1711},[123,127],[57,1713,1715],{"className":1714},[132],[57,1716,1718,1738],{"className":1717},[136,137],[57,1719,1721,1735],{"className":1720},[141],[57,1722,1724],{"className":1723,"style":339},[145],[57,1725,1726,1729],{"style":550},[57,1727],{"className":1728,"style":154},[153],[57,1730,1732],{"className":1731},[158,159,160,161],[57,1733,280],{"className":1734},[123,161],[57,1736,169],{"className":1737},[168],[57,1739,1741],{"className":1740},[141],[57,1742,1744],{"className":1743,"style":176},[145],[57,1745],{},[57,1747,283],{"className":1748},[365],[57,1750],{"className":1751,"style":119},[118],[57,1753,1755,1758],{"className":1754},[123],[57,1756,1015],{"className":1757},[123,127],[57,1759,1761],{"className":1760},[132],[57,1762,1764,1784],{"className":1763},[136,137],[57,1765,1767,1781],{"className":1766},[141],[57,1768,1770],{"className":1769,"style":339},[145],[57,1771,1772,1775],{"style":550},[57,1773],{"className":1774,"style":154},[153],[57,1776,1778],{"className":1777},[158,159,160,161],[57,1779,290],{"className":1780},[123,161],[57,1782,169],{"className":1783},[168],[57,1785,1787],{"className":1786},[141],[57,1788,1790],{"className":1789,"style":176},[145],[57,1791],{},[57,1793,513],{"className":1794},[679],[57,1796],{"className":1797,"style":1281},[118],[57,1799,1030],{"className":1800},[1285],[57,1802],{"className":1803,"style":1281},[118],[57,1805,1807,1810,1813,1853,1893,1896,1899,1939,1979],{"className":1806},[102],[57,1808],{"className":1809,"style":107},[106],[57,1811,478],{"className":1812},[529],[57,1814,1816,1819],{"className":1815},[123],[57,1817,483],{"className":1818},[123,127],[57,1820,1822],{"className":1821},[132],[57,1823,1825,1845],{"className":1824},[136,137],[57,1826,1828,1842],{"className":1827},[141],[57,1829,1831],{"className":1830,"style":339},[145],[57,1832,1833,1836],{"style":550},[57,1834],{"className":1835,"style":154},[153],[57,1837,1839],{"className":1838},[158,159,160,161],[57,1840,280],{"className":1841},[123,161],[57,1843,169],{"className":1844},[168],[57,1846,1848],{"className":1847},[141],[57,1849,1851],{"className":1850,"style":176},[145],[57,1852],{},[57,1854,1856,1859],{"className":1855},[123],[57,1857,1015],{"className":1858},[123,127],[57,1860,1862],{"className":1861},[132],[57,1863,1865,1885],{"className":1864},[136,137],[57,1866,1868,1882],{"className":1867},[141],[57,1869,1871],{"className":1870,"style":339},[145],[57,1872,1873,1876],{"style":550},[57,1874],{"className":1875,"style":154},[153],[57,1877,1879],{"className":1878},[158,159,160,161],[57,1880,280],{"className":1881},[123,161],[57,1883,169],{"className":1884},[168],[57,1886,1888],{"className":1887},[141],[57,1889,1891],{"className":1890,"style":176},[145],[57,1892],{},[57,1894,283],{"className":1895},[365],[57,1897],{"className":1898,"style":119},[118],[57,1900,1902,1905],{"className":1901},[123],[57,1903,483],{"className":1904},[123,127],[57,1906,1908],{"className":1907},[132],[57,1909,1911,1931],{"className":1910},[136,137],[57,1912,1914,1928],{"className":1913},[141],[57,1915,1917],{"className":1916,"style":339},[145],[57,1918,1919,1922],{"style":550},[57,1920],{"className":1921,"style":154},[153],[57,1923,1925],{"className":1924},[158,159,160,161],[57,1926,290],{"className":1927},[123,161],[57,1929,169],{"className":1930},[168],[57,1932,1934],{"className":1933},[141],[57,1935,1937],{"className":1936,"style":176},[145],[57,1938],{},[57,1940,1942,1945],{"className":1941},[123],[57,1943,1015],{"className":1944},[123,127],[57,1946,1948],{"className":1947},[132],[57,1949,1951,1971],{"className":1950},[136,137],[57,1952,1954,1968],{"className":1953},[141],[57,1955,1957],{"className":1956,"style":339},[145],[57,1958,1959,1962],{"style":550},[57,1960],{"className":1961,"style":154},[153],[57,1963,1965],{"className":1964},[158,159,160,161],[57,1966,290],{"className":1967},[123,161],[57,1969,169],{"className":1970},[168],[57,1972,1974],{"className":1973},[141],[57,1975,1977],{"className":1976,"style":176},[145],[57,1978],{},[57,1980,513],{"className":1981},[679],[46,1983,1985],{"id":1984},"_2-定义-341ii直和-direct-sum","2. 定义 3.4.1(ii)：直和 (Direct Sum)",[14,1987,1988,55,1990,2069,2070,2149,55],{},[24,1989,54],{},[57,1991,1993,2014],{"className":1992},[60],[57,1994,1996],{"className":1995},[64],[66,1997,1998],{"xmlns":68},[70,1999,2000,2011],{},[73,2001,2002,2005],{},[76,2003,2004],{},"∐",[80,2006,2007,2009],{},[83,2008,85],{},[83,2010,88],{},[90,2012,2013],{"encoding":92},"\\coprod R_i",[57,2015,2017],{"className":2016,"ariaHidden":98},[97],[57,2018,2020,2023,2026,2029],{"className":2019},[102],[57,2021],{"className":2022,"style":107},[106],[57,2024,2004],{"className":2025,"style":114},[111,112,113],[57,2027],{"className":2028,"style":119},[118],[57,2030,2032,2035],{"className":2031},[123],[57,2033,85],{"className":2034,"style":128},[123,127],[57,2036,2038],{"className":2037},[132],[57,2039,2041,2061],{"className":2040},[136,137],[57,2042,2044,2058],{"className":2043},[141],[57,2045,2047],{"className":2046,"style":146},[145],[57,2048,2049,2052],{"style":149},[57,2050],{"className":2051,"style":154},[153],[57,2053,2055],{"className":2054},[158,159,160,161],[57,2056,88],{"className":2057},[123,127,161],[57,2059,169],{"className":2060},[168],[57,2062,2064],{"className":2063},[141],[57,2065,2067],{"className":2066,"style":176},[145],[57,2068],{}," 或 ",[57,2071,2073,2094],{"className":2072},[60],[57,2074,2076],{"className":2075},[64],[66,2077,2078],{"xmlns":68},[70,2079,2080,2091],{},[73,2081,2082,2085],{},[76,2083,2084],{},"⨁",[80,2086,2087,2089],{},[83,2088,85],{},[83,2090,88],{},[90,2092,2093],{"encoding":92},"\\bigoplus R_i",[57,2095,2097],{"className":2096,"ariaHidden":98},[97],[57,2098,2100,2103,2106,2109],{"className":2099},[102],[57,2101],{"className":2102,"style":107},[106],[57,2104,2084],{"className":2105,"style":114},[111,112,113],[57,2107],{"className":2108,"style":119},[118],[57,2110,2112,2115],{"className":2111},[123],[57,2113,85],{"className":2114,"style":128},[123,127],[57,2116,2118],{"className":2117},[132],[57,2119,2121,2141],{"className":2120},[136,137],[57,2122,2124,2138],{"className":2123},[141],[57,2125,2127],{"className":2126,"style":146},[145],[57,2128,2129,2132],{"style":149},[57,2130],{"className":2131,"style":154},[153],[57,2133,2135],{"className":2134},[158,159,160,161],[57,2136,88],{"className":2137},[123,127,161],[57,2139,169],{"className":2140},[168],[57,2142,2144],{"className":2143},[141],[57,2145,2147],{"className":2146,"style":176},[145],[57,2148],{},[24,2150,2151],{},"它和直积的区别",[184,2153,2154,2160,2170],{},[21,2155,2156,2159],{},[24,2157,2158],{},"直积","：列表是无限长的，每个位置都可以是非零数。",[21,2161,2162,2165,2166,2169],{},[24,2163,2164],{},"直和","：列表虽然可能无限长，但要求",[24,2167,2168],{},"只有有限个位置是非零的","，剩下的全是 0。",[21,2171,2172,2175,2176,2318,2319,2322],{},[24,2173,2174],{},"特别注意","：如果只有有限个环（比如 ",[57,2177,2179,2210],{"className":2178},[60],[57,2180,2182],{"className":2181},[64],[66,2183,2184],{"xmlns":68},[70,2185,2186,2207],{},[73,2187,2188,2194,2196,2198,2200],{},[80,2189,2190,2192],{},[83,2191,85],{},[278,2193,280],{},[76,2195,283],{"separator":98},[76,2197,506],{},[76,2199,283],{"separator":98},[80,2201,2202,2204],{},[83,2203,85],{},[83,2205,2206],{},"n",[90,2208,2209],{"encoding":92},"R_1, \\dots, R_n",[57,2211,2213],{"className":2212,"ariaHidden":98},[97],[57,2214,2216,2219,2259,2262,2265,2268,2271,2274,2277],{"className":2215},[102],[57,2217],{"className":2218,"style":320},[106],[57,2220,2222,2225],{"className":2221},[123],[57,2223,85],{"className":2224,"style":128},[123,127],[57,2226,2228],{"className":2227},[132],[57,2229,2231,2251],{"className":2230},[136,137],[57,2232,2234,2248],{"className":2233},[141],[57,2235,2237],{"className":2236,"style":339},[145],[57,2238,2239,2242],{"style":149},[57,2240],{"className":2241,"style":154},[153],[57,2243,2245],{"className":2244},[158,159,160,161],[57,2246,280],{"className":2247},[123,161],[57,2249,169],{"className":2250},[168],[57,2252,2254],{"className":2253},[141],[57,2255,2257],{"className":2256,"style":176},[145],[57,2258],{},[57,2260,283],{"className":2261},[365],[57,2263],{"className":2264,"style":119},[118],[57,2266,506],{"className":2267},[672],[57,2269],{"className":2270,"style":119},[118],[57,2272,283],{"className":2273},[365],[57,2275],{"className":2276,"style":119},[118],[57,2278,2280,2283],{"className":2279},[123],[57,2281,85],{"className":2282,"style":128},[123,127],[57,2284,2286],{"className":2285},[132],[57,2287,2289,2310],{"className":2288},[136,137],[57,2290,2292,2307],{"className":2291},[141],[57,2293,2296],{"className":2294,"style":2295},[145],"height:0.1514em;",[57,2297,2298,2301],{"style":149},[57,2299],{"className":2300,"style":154},[153],[57,2302,2304],{"className":2303},[158,159,160,161],[57,2305,2206],{"className":2306},[123,127,161],[57,2308,169],{"className":2309},[168],[57,2311,2313],{"className":2312},[141],[57,2314,2316],{"className":2315,"style":176},[145],[57,2317],{},"），那么",[24,2320,2321],{},"直积 = 直和","。这就是为什么后面的定理里，这两个符号经常混用，因为我们处理的通常是有限个理想的情况。",[35,2324],{},[38,2326,2328],{"id":2327},"第二阶段重头戏中国剩余定理证明详解","第二阶段：重头戏——中国剩余定理（证明详解）",[14,2330,2331],{},"这是最难的部分。",[14,2333,2334,2337,2338,2481,2482,2511,2512,2515,2516],{},[24,2335,2336],{},"定理 3.4.2 (中国剩余定理 CRT)","\n设 ",[57,2339,2341,2372],{"className":2340},[60],[57,2342,2344],{"className":2343},[64],[66,2345,2346],{"xmlns":68},[70,2347,2348,2369],{},[73,2349,2350,2357,2359,2361,2363],{},[80,2351,2352,2355],{},[83,2353,2354],{},"I",[278,2356,280],{},[76,2358,283],{"separator":98},[76,2360,506],{},[76,2362,283],{"separator":98},[80,2364,2365,2367],{},[83,2366,2354],{},[83,2368,2206],{},[90,2370,2371],{"encoding":92},"I_1, \\dots, I_n",[57,2373,2375],{"className":2374,"ariaHidden":98},[97],[57,2376,2378,2381,2423,2426,2429,2432,2435,2438,2441],{"className":2377},[102],[57,2379],{"className":2380,"style":320},[106],[57,2382,2384,2388],{"className":2383},[123],[57,2385,2354],{"className":2386,"style":2387},[123,127],"margin-right:0.0785em;",[57,2389,2391],{"className":2390},[132],[57,2392,2394,2415],{"className":2393},[136,137],[57,2395,2397,2412],{"className":2396},[141],[57,2398,2400],{"className":2399,"style":339},[145],[57,2401,2403,2406],{"style":2402},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[57,2404],{"className":2405,"style":154},[153],[57,2407,2409],{"className":2408},[158,159,160,161],[57,2410,280],{"className":2411},[123,161],[57,2413,169],{"className":2414},[168],[57,2416,2418],{"className":2417},[141],[57,2419,2421],{"className":2420,"style":176},[145],[57,2422],{},[57,2424,283],{"className":2425},[365],[57,2427],{"className":2428,"style":119},[118],[57,2430,506],{"className":2431},[672],[57,2433],{"className":2434,"style":119},[118],[57,2436,283],{"className":2437},[365],[57,2439],{"className":2440,"style":119},[118],[57,2442,2444,2447],{"className":2443},[123],[57,2445,2354],{"className":2446,"style":2387},[123,127],[57,2448,2450],{"className":2449},[132],[57,2451,2453,2473],{"className":2452},[136,137],[57,2454,2456,2470],{"className":2455},[141],[57,2457,2459],{"className":2458,"style":2295},[145],[57,2460,2461,2464],{"style":2402},[57,2462],{"className":2463,"style":154},[153],[57,2465,2467],{"className":2466},[158,159,160,161],[57,2468,2206],{"className":2469},[123,127,161],[57,2471,169],{"className":2472},[168],[57,2474,2476],{"className":2475},[141],[57,2477,2479],{"className":2478,"style":176},[145],[57,2480],{}," 是环 ",[57,2483,2485,2498],{"className":2484},[60],[57,2486,2488],{"className":2487},[64],[66,2489,2490],{"xmlns":68},[70,2491,2492,2496],{},[73,2493,2494],{},[83,2495,85],{},[90,2497,85],{"encoding":92},[57,2499,2501],{"className":2500,"ariaHidden":98},[97],[57,2502,2504,2508],{"className":2503},[102],[57,2505],{"className":2506,"style":2507},[106],"height:0.6833em;",[57,2509,85],{"className":2510,"style":128},[123,127]," 的理想，且",[24,2513,2514],{},"两两互素","。则：\n",[57,2517,2519,2590],{"className":2518},[60],[57,2520,2522],{"className":2521},[64],[66,2523,2524],{"xmlns":68},[70,2525,2526,2587],{},[73,2527,2528,2530,2533,2535,2552,2558,2560,2563,2577,2579,2581],{},[83,2529,85],{},[83,2531,2532],{"mathvariant":302},"\u002F",[76,2534,478],{"stretchy":477},[2536,2537,2538,2541,2550],"msubsup",{},[76,2539,2540],{},"⋂",[73,2542,2543,2546,2548],{},[83,2544,2545],{},"k",[76,2547,1030],{},[278,2549,280],{},[83,2551,2206],{},[80,2553,2554,2556],{},[83,2555,2354],{},[83,2557,2545],{},[76,2559,513],{"stretchy":477},[76,2561,2562],{},"≅",[2536,2564,2565,2567,2575],{},[76,2566,2084],{},[73,2568,2569,2571,2573],{},[83,2570,2545],{},[76,2572,1030],{},[278,2574,280],{},[83,2576,2206],{},[83,2578,85],{},[83,2580,2532],{"mathvariant":302},[80,2582,2583,2585],{},[83,2584,2354],{},[83,2586,2545],{},[90,2588,2589],{"encoding":92},"R \u002F (\\bigcap_{k=1}^n I_k) \\cong \\bigoplus_{k=1}^n R\u002FI_k",[57,2591,2593,2730],{"className":2592,"ariaHidden":98},[97],[57,2594,2596,2600,2603,2606,2609,2674,2677,2718,2721,2724,2727],{"className":2595},[102],[57,2597],{"className":2598,"style":2599},[106],"height:1.104em;vertical-align:-0.2997em;",[57,2601,85],{"className":2602,"style":128},[123,127],[57,2604,2532],{"className":2605},[123],[57,2607,478],{"className":2608},[529],[57,2610,2612,2615],{"className":2611},[111],[57,2613,2540],{"className":2614,"style":114},[111,112,113],[57,2616,2618],{"className":2617},[132],[57,2619,2621,2665],{"className":2620},[136,137],[57,2622,2624,2662],{"className":2623},[141],[57,2625,2628,2650],{"className":2626,"style":2627},[145],"height:0.8043em;",[57,2629,2631,2634],{"style":2630},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[57,2632],{"className":2633,"style":154},[153],[57,2635,2637],{"className":2636},[158,159,160,161],[57,2638,2640,2644,2647],{"className":2639},[123,161],[57,2641,2545],{"className":2642,"style":2643},[123,127,161],"margin-right:0.0315em;",[57,2645,1030],{"className":2646},[1285,161],[57,2648,280],{"className":2649},[123,161],[57,2651,2653,2656],{"style":2652},"top:-3.2029em;margin-right:0.05em;",[57,2654],{"className":2655,"style":154},[153],[57,2657,2659],{"className":2658},[158,159,160,161],[57,2660,2206],{"className":2661},[123,127,161],[57,2663,169],{"className":2664},[168],[57,2666,2668],{"className":2667},[141],[57,2669,2672],{"className":2670,"style":2671},[145],"height:0.2997em;",[57,2673],{},[57,2675],{"className":2676,"style":119},[118],[57,2678,2680,2683],{"className":2679},[123],[57,2681,2354],{"className":2682,"style":2387},[123,127],[57,2684,2686],{"className":2685},[132],[57,2687,2689,2710],{"className":2688},[136,137],[57,2690,2692,2707],{"className":2691},[141],[57,2693,2696],{"className":2694,"style":2695},[145],"height:0.3361em;",[57,2697,2698,2701],{"style":2402},[57,2699],{"className":2700,"style":154},[153],[57,2702,2704],{"className":2703},[158,159,160,161],[57,2705,2545],{"className":2706,"style":2643},[123,127,161],[57,2708,169],{"className":2709},[168],[57,2711,2713],{"className":2712},[141],[57,2714,2716],{"className":2715,"style":176},[145],[57,2717],{},[57,2719,513],{"className":2720},[679],[57,2722],{"className":2723,"style":1281},[118],[57,2725,2562],{"className":2726},[1285],[57,2728],{"className":2729,"style":1281},[118],[57,2731,2733,2736,2796,2799,2802,2805],{"className":2732},[102],[57,2734],{"className":2735,"style":2599},[106],[57,2737,2739,2742],{"className":2738},[111],[57,2740,2084],{"className":2741,"style":114},[111,112,113],[57,2743,2745],{"className":2744},[132],[57,2746,2748,2788],{"className":2747},[136,137],[57,2749,2751,2785],{"className":2750},[141],[57,2752,2754,2774],{"className":2753,"style":2627},[145],[57,2755,2756,2759],{"style":2630},[57,2757],{"className":2758,"style":154},[153],[57,2760,2762],{"className":2761},[158,159,160,161],[57,2763,2765,2768,2771],{"className":2764},[123,161],[57,2766,2545],{"className":2767,"style":2643},[123,127,161],[57,2769,1030],{"className":2770},[1285,161],[57,2772,280],{"className":2773},[123,161],[57,2775,2776,2779],{"style":2652},[57,2777],{"className":2778,"style":154},[153],[57,2780,2782],{"className":2781},[158,159,160,161],[57,2783,2206],{"className":2784},[123,127,161],[57,2786,169],{"className":2787},[168],[57,2789,2791],{"className":2790},[141],[57,2792,2794],{"className":2793,"style":2671},[145],[57,2795],{},[57,2797],{"className":2798,"style":119},[118],[57,2800,85],{"className":2801,"style":128},[123,127],[57,2803,2532],{"className":2804},[123],[57,2806,2808,2811],{"className":2807},[123],[57,2809,2354],{"className":2810,"style":2387},[123,127],[57,2812,2814],{"className":2813},[132],[57,2815,2817,2837],{"className":2816},[136,137],[57,2818,2820,2834],{"className":2819},[141],[57,2821,2823],{"className":2822,"style":2695},[145],[57,2824,2825,2828],{"style":2402},[57,2826],{"className":2827,"style":154},[153],[57,2829,2831],{"className":2830},[158,159,160,161],[57,2832,2545],{"className":2833,"style":2643},[123,127,161],[57,2835,169],{"className":2836},[168],[57,2838,2840],{"className":2839},[141],[57,2841,2843],{"className":2842,"style":176},[145],[57,2844],{},[14,2846,2847,2850,2851,2879,2880,2910,2911,2985,2986,2879,3040,3094,3095,3171],{},[24,2848,2849],{},"前提知识补充：什么是“两两互素”？","\n在环论里，理想 ",[57,2852,2854,2867],{"className":2853},[60],[57,2855,2857],{"className":2856},[64],[66,2858,2859],{"xmlns":68},[70,2860,2861,2865],{},[73,2862,2863],{},[83,2864,2354],{},[90,2866,2354],{"encoding":92},[57,2868,2870],{"className":2869,"ariaHidden":98},[97],[57,2871,2873,2876],{"className":2872},[102],[57,2874],{"className":2875,"style":2507},[106],[57,2877,2354],{"className":2878,"style":2387},[123,127]," 和 ",[57,2881,2883,2897],{"className":2882},[60],[57,2884,2886],{"className":2885},[64],[66,2887,2888],{"xmlns":68},[70,2889,2890,2895],{},[73,2891,2892],{},[83,2893,2894],{},"J",[90,2896,2894],{"encoding":92},[57,2898,2900],{"className":2899,"ariaHidden":98},[97],[57,2901,2903,2906],{"className":2902},[102],[57,2904],{"className":2905,"style":2507},[106],[57,2907,2894],{"className":2908,"style":2909},[123,127],"margin-right:0.0962em;"," 互素意味着 ",[57,2912,2914,2936],{"className":2913},[60],[57,2915,2917],{"className":2916},[64],[66,2918,2919],{"xmlns":68},[70,2920,2921,2933],{},[73,2922,2923,2925,2927,2929,2931],{},[83,2924,2354],{},[76,2926,1008],{},[83,2928,2894],{},[76,2930,1030],{},[83,2932,85],{},[90,2934,2935],{"encoding":92},"I + J = R",[57,2937,2939,2958,2976],{"className":2938,"ariaHidden":98},[97],[57,2940,2942,2946,2949,2952,2955],{"className":2941},[102],[57,2943],{"className":2944,"style":2945},[106],"height:0.7667em;vertical-align:-0.0833em;",[57,2947,2354],{"className":2948,"style":2387},[123,127],[57,2950],{"className":2951,"style":1172},[118],[57,2953,1008],{"className":2954},[1176],[57,2956],{"className":2957,"style":1172},[118],[57,2959,2961,2964,2967,2970,2973],{"className":2960},[102],[57,2962],{"className":2963,"style":2507},[106],[57,2965,2894],{"className":2966,"style":2909},[123,127],[57,2968],{"className":2969,"style":1281},[118],[57,2971,1030],{"className":2972},[1285],[57,2974],{"className":2975,"style":1281},[118],[57,2977,2979,2982],{"className":2978},[102],[57,2980],{"className":2981,"style":2507},[106],[57,2983,85],{"className":2984,"style":128},[123,127],"。\n这意味着，我们可以找到 ",[57,2987,2989,3009],{"className":2988},[60],[57,2990,2992],{"className":2991},[64],[66,2993,2994],{"xmlns":68},[70,2995,2996,3006],{},[73,2997,2998,3001,3004],{},[83,2999,3000],{},"x",[76,3002,3003],{},"∈",[83,3005,2354],{},[90,3007,3008],{"encoding":92},"x \\in I",[57,3010,3012,3031],{"className":3011,"ariaHidden":98},[97],[57,3013,3015,3019,3022,3025,3028],{"className":3014},[102],[57,3016],{"className":3017,"style":3018},[106],"height:0.5782em;vertical-align:-0.0391em;",[57,3020,3000],{"className":3021},[123,127],[57,3023],{"className":3024,"style":1281},[118],[57,3026,3003],{"className":3027},[1285],[57,3029],{"className":3030,"style":1281},[118],[57,3032,3034,3037],{"className":3033},[102],[57,3035],{"className":3036,"style":2507},[106],[57,3038,2354],{"className":3039,"style":2387},[123,127],[57,3041,3043,3062],{"className":3042},[60],[57,3044,3046],{"className":3045},[64],[66,3047,3048],{"xmlns":68},[70,3049,3050,3059],{},[73,3051,3052,3055,3057],{},[83,3053,3054],{},"y",[76,3056,3003],{},[83,3058,2894],{},[90,3060,3061],{"encoding":92},"y \\in J",[57,3063,3065,3085],{"className":3064,"ariaHidden":98},[97],[57,3066,3068,3072,3076,3079,3082],{"className":3067},[102],[57,3069],{"className":3070,"style":3071},[106],"height:0.7335em;vertical-align:-0.1944em;",[57,3073,3054],{"className":3074,"style":3075},[123,127],"margin-right:0.0359em;",[57,3077],{"className":3078,"style":1281},[118],[57,3080,3003],{"className":3081},[1285],[57,3083],{"className":3084,"style":1281},[118],[57,3086,3088,3091],{"className":3087},[102],[57,3089],{"className":3090,"style":2507},[106],[57,3092,2894],{"className":3093,"style":2909},[123,127],"，使得 ",[57,3096,3098,3120],{"className":3097},[60],[57,3099,3101],{"className":3100},[64],[66,3102,3103],{"xmlns":68},[70,3104,3105,3117],{},[73,3106,3107,3109,3111,3113,3115],{},[83,3108,3000],{},[76,3110,1008],{},[83,3112,3054],{},[76,3114,1030],{},[278,3116,280],{},[90,3118,3119],{"encoding":92},"x + y = 1",[57,3121,3123,3142,3161],{"className":3122,"ariaHidden":98},[97],[57,3124,3126,3130,3133,3136,3139],{"className":3125},[102],[57,3127],{"className":3128,"style":3129},[106],"height:0.6667em;vertical-align:-0.0833em;",[57,3131,3000],{"className":3132},[123,127],[57,3134],{"className":3135,"style":1172},[118],[57,3137,1008],{"className":3138},[1176],[57,3140],{"className":3141,"style":1172},[118],[57,3143,3145,3149,3152,3155,3158],{"className":3144},[102],[57,3146],{"className":3147,"style":3148},[106],"height:0.625em;vertical-align:-0.1944em;",[57,3150,3054],{"className":3151,"style":3075},[123,127],[57,3153],{"className":3154,"style":1281},[118],[57,3156,1030],{"className":3157},[1285],[57,3159],{"className":3160,"style":1281},[118],[57,3162,3164,3168],{"className":3163},[102],[57,3165],{"className":3166,"style":3167},[106],"height:0.6444em;",[57,3169,280],{"className":3170},[123],"。这一点在后面的证明中至关重要（用来制造“1”）。",[46,3173,3175],{"id":3174},"证明第一步构造映射","证明第一步：构造映射",[14,3177,3178],{},"我们想证明左边（大环模交集）和右边（小环直和）是一样的（同构），首先得架一座桥。",[14,3180,3181,3182,3317],{},"定义映射 ",[57,3183,3185,3220],{"className":3184},[60],[57,3186,3188],{"className":3187},[64],[66,3189,3190],{"xmlns":68},[70,3191,3192,3217],{},[73,3193,3194,3197,3200,3202,3205,3207,3209,3211],{},[83,3195,3196],{},"ϕ",[76,3198,3199],{},":",[83,3201,85],{},[76,3203,3204],{},"→",[76,3206,2084],{},[83,3208,85],{},[83,3210,2532],{"mathvariant":302},[80,3212,3213,3215],{},[83,3214,2354],{},[83,3216,2545],{},[90,3218,3219],{"encoding":92},"\\phi: R \\to \\bigoplus R\u002FI_k",[57,3221,3223,3241,3259],{"className":3222,"ariaHidden":98},[97],[57,3224,3226,3229,3232,3235,3238],{"className":3225},[102],[57,3227],{"className":3228,"style":1353},[106],[57,3230,3196],{"className":3231},[123,127],[57,3233],{"className":3234,"style":1281},[118],[57,3236,3199],{"className":3237},[1285],[57,3239],{"className":3240,"style":1281},[118],[57,3242,3244,3247,3250,3253,3256],{"className":3243},[102],[57,3245],{"className":3246,"style":2507},[106],[57,3248,85],{"className":3249,"style":128},[123,127],[57,3251],{"className":3252,"style":1281},[118],[57,3254,3204],{"className":3255},[1285],[57,3257],{"className":3258,"style":1281},[118],[57,3260,3262,3265,3268,3271,3274,3277],{"className":3261},[102],[57,3263],{"className":3264,"style":107},[106],[57,3266,2084],{"className":3267,"style":114},[111,112,113],[57,3269],{"className":3270,"style":119},[118],[57,3272,85],{"className":3273,"style":128},[123,127],[57,3275,2532],{"className":3276},[123],[57,3278,3280,3283],{"className":3279},[123],[57,3281,2354],{"className":3282,"style":2387},[123,127],[57,3284,3286],{"className":3285},[132],[57,3287,3289,3309],{"className":3288},[136,137],[57,3290,3292,3306],{"className":3291},[141],[57,3293,3295],{"className":3294,"style":2695},[145],[57,3296,3297,3300],{"style":2402},[57,3298],{"className":3299,"style":154},[153],[57,3301,3303],{"className":3302},[158,159,160,161],[57,3304,2545],{"className":3305,"style":2643},[123,127,161],[57,3307,169],{"className":3308},[168],[57,3310,3312],{"className":3311},[141],[57,3313,3315],{"className":3314,"style":176},[145],[57,3316],{},[57,3318,3320,3384],{"className":3319},[60],[57,3321,3323],{"className":3322},[64],[66,3324,3325],{"xmlns":68},[70,3326,3327,3381],{},[73,3328,3329,3331,3333,3335,3337,3339,3341,3343,3345,3351,3353,3355,3357,3363,3365,3367,3369,3371,3373,3379],{},[83,3330,3196],{},[76,3332,478],{"stretchy":477},[83,3334,483],{},[76,3336,513],{"stretchy":477},[76,3338,1030],{},[76,3340,478],{"stretchy":477},[83,3342,483],{},[76,3344,1008],{},[80,3346,3347,3349],{},[83,3348,2354],{},[278,3350,280],{},[76,3352,283],{"separator":98},[83,3354,483],{},[76,3356,1008],{},[80,3358,3359,3361],{},[83,3360,2354],{},[278,3362,290],{},[76,3364,283],{"separator":98},[76,3366,506],{},[76,3368,283],{"separator":98},[83,3370,483],{},[76,3372,1008],{},[80,3374,3375,3377],{},[83,3376,2354],{},[83,3378,2206],{},[76,3380,513],{"stretchy":477},[90,3382,3383],{"encoding":92},"\\phi(a) = (a + I_1, a + I_2, \\dots, a + I_n)",[57,3385,3387,3414,3435,3499,3575],{"className":3386,"ariaHidden":98},[97],[57,3388,3390,3393,3396,3399,3402,3405,3408,3411],{"className":3389},[102],[57,3391],{"className":3392,"style":107},[106],[57,3394,3196],{"className":3395},[123,127],[57,3397,478],{"className":3398},[529],[57,3400,483],{"className":3401},[123,127],[57,3403,513],{"className":3404},[679],[57,3406],{"className":3407,"style":1281},[118],[57,3409,1030],{"className":3410},[1285],[57,3412],{"className":3413,"style":1281},[118],[57,3415,3417,3420,3423,3426,3429,3432],{"className":3416},[102],[57,3418],{"className":3419,"style":107},[106],[57,3421,478],{"className":3422},[529],[57,3424,483],{"className":3425},[123,127],[57,3427],{"className":3428,"style":1172},[118],[57,3430,1008],{"className":3431},[1176],[57,3433],{"className":3434,"style":1172},[118],[57,3436,3438,3441,3481,3484,3487,3490,3493,3496],{"className":3437},[102],[57,3439],{"className":3440,"style":320},[106],[57,3442,3444,3447],{"className":3443},[123],[57,3445,2354],{"className":3446,"style":2387},[123,127],[57,3448,3450],{"className":3449},[132],[57,3451,3453,3473],{"className":3452},[136,137],[57,3454,3456,3470],{"className":3455},[141],[57,3457,3459],{"className":3458,"style":339},[145],[57,3460,3461,3464],{"style":2402},[57,3462],{"className":3463,"style":154},[153],[57,3465,3467],{"className":3466},[158,159,160,161],[57,3468,280],{"className":3469},[123,161],[57,3471,169],{"className":3472},[168],[57,3474,3476],{"className":3475},[141],[57,3477,3479],{"className":3478,"style":176},[145],[57,3480],{},[57,3482,283],{"className":3483},[365],[57,3485],{"className":3486,"style":119},[118],[57,3488,483],{"className":3489},[123,127],[57,3491],{"className":3492,"style":1172},[118],[57,3494,1008],{"className":3495},[1176],[57,3497],{"className":3498,"style":1172},[118],[57,3500,3502,3505,3545,3548,3551,3554,3557,3560,3563,3566,3569,3572],{"className":3501},[102],[57,3503],{"className":3504,"style":320},[106],[57,3506,3508,3511],{"className":3507},[123],[57,3509,2354],{"className":3510,"style":2387},[123,127],[57,3512,3514],{"className":3513},[132],[57,3515,3517,3537],{"className":3516},[136,137],[57,3518,3520,3534],{"className":3519},[141],[57,3521,3523],{"className":3522,"style":339},[145],[57,3524,3525,3528],{"style":2402},[57,3526],{"className":3527,"style":154},[153],[57,3529,3531],{"className":3530},[158,159,160,161],[57,3532,290],{"className":3533},[123,161],[57,3535,169],{"className":3536},[168],[57,3538,3540],{"className":3539},[141],[57,3541,3543],{"className":3542,"style":176},[145],[57,3544],{},[57,3546,283],{"className":3547},[365],[57,3549],{"className":3550,"style":119},[118],[57,3552,506],{"className":3553},[672],[57,3555],{"className":3556,"style":119},[118],[57,3558,283],{"className":3559},[365],[57,3561],{"className":3562,"style":119},[118],[57,3564,483],{"className":3565},[123,127],[57,3567],{"className":3568,"style":1172},[118],[57,3570,1008],{"className":3571},[1176],[57,3573],{"className":3574,"style":1172},[118],[57,3576,3578,3581,3621],{"className":3577},[102],[57,3579],{"className":3580,"style":107},[106],[57,3582,3584,3587],{"className":3583},[123],[57,3585,2354],{"className":3586,"style":2387},[123,127],[57,3588,3590],{"className":3589},[132],[57,3591,3593,3613],{"className":3592},[136,137],[57,3594,3596,3610],{"className":3595},[141],[57,3597,3599],{"className":3598,"style":2295},[145],[57,3600,3601,3604],{"style":2402},[57,3602],{"className":3603,"style":154},[153],[57,3605,3607],{"className":3606},[158,159,160,161],[57,3608,2206],{"className":3609},[123,127,161],[57,3611,169],{"className":3612},[168],[57,3614,3616],{"className":3615},[141],[57,3617,3619],{"className":3618,"style":176},[145],[57,3620],{},[57,3622,513],{"className":3623},[679],[184,3625,3626,3824],{},[21,3627,3628,3631,3632,3660,3661,3690,3691,3823],{},[24,3629,3630],{},"解释","：把 ",[57,3633,3635,3648],{"className":3634},[60],[57,3636,3638],{"className":3637},[64],[66,3639,3640],{"xmlns":68},[70,3641,3642,3646],{},[73,3643,3644],{},[83,3645,85],{},[90,3647,85],{"encoding":92},[57,3649,3651],{"className":3650,"ariaHidden":98},[97],[57,3652,3654,3657],{"className":3653},[102],[57,3655],{"className":3656,"style":2507},[106],[57,3658,85],{"className":3659,"style":128},[123,127]," 里的一个元素 ",[57,3662,3664,3677],{"className":3663},[60],[57,3665,3667],{"className":3666},[64],[66,3668,3669],{"xmlns":68},[70,3670,3671,3675],{},[73,3672,3673],{},[83,3674,483],{},[90,3676,483],{"encoding":92},[57,3678,3680],{"className":3679,"ariaHidden":98},[97],[57,3681,3683,3687],{"className":3682},[102],[57,3684],{"className":3685,"style":3686},[106],"height:0.4306em;",[57,3688,483],{"className":3689},[123,127],"，分别拿到 ",[57,3692,3694,3722],{"className":3693},[60],[57,3695,3697],{"className":3696},[64],[66,3698,3699],{"xmlns":68},[70,3700,3701,3719],{},[73,3702,3703,3709,3711,3717],{},[80,3704,3705,3707],{},[83,3706,2354],{},[278,3708,280],{},[76,3710,283],{"separator":98},[80,3712,3713,3715],{},[83,3714,2354],{},[278,3716,290],{},[76,3718,506],{},[90,3720,3721],{"encoding":92},"I_1, I_2 \\dots",[57,3723,3725],{"className":3724,"ariaHidden":98},[97],[57,3726,3728,3731,3771,3774,3777,3817,3820],{"className":3727},[102],[57,3729],{"className":3730,"style":320},[106],[57,3732,3734,3737],{"className":3733},[123],[57,3735,2354],{"className":3736,"style":2387},[123,127],[57,3738,3740],{"className":3739},[132],[57,3741,3743,3763],{"className":3742},[136,137],[57,3744,3746,3760],{"className":3745},[141],[57,3747,3749],{"className":3748,"style":339},[145],[57,3750,3751,3754],{"style":2402},[57,3752],{"className":3753,"style":154},[153],[57,3755,3757],{"className":3756},[158,159,160,161],[57,3758,280],{"className":3759},[123,161],[57,3761,169],{"className":3762},[168],[57,3764,3766],{"className":3765},[141],[57,3767,3769],{"className":3768,"style":176},[145],[57,3770],{},[57,3772,283],{"className":3773},[365],[57,3775],{"className":3776,"style":119},[118],[57,3778,3780,3783],{"className":3779},[123],[57,3781,2354],{"className":3782,"style":2387},[123,127],[57,3784,3786],{"className":3785},[132],[57,3787,3789,3809],{"className":3788},[136,137],[57,3790,3792,3806],{"className":3791},[141],[57,3793,3795],{"className":3794,"style":339},[145],[57,3796,3797,3800],{"style":2402},[57,3798],{"className":3799,"style":154},[153],[57,3801,3803],{"className":3802},[158,159,160,161],[57,3804,290],{"className":3805},[123,161],[57,3807,169],{"className":3808},[168],[57,3810,3812],{"className":3811},[141],[57,3813,3815],{"className":3814,"style":176},[145],[57,3816],{},[57,3818],{"className":3819,"style":119},[118],[57,3821,506],{"className":3822},[672]," 里去取余数，组成一个列表。",[21,3825,3826],{},"显然这是一个同态（保持加法和乘法）。",[14,3828,3829],{},[24,3830,3831],{},"我们要找它的核 (Kernel)：",[184,3833,3834,3907,4215,4441],{},[21,3835,3836,3877,3878,3906],{},[57,3837,3839,3859],{"className":3838},[60],[57,3840,3842],{"className":3841},[64],[66,3843,3844],{"xmlns":68},[70,3845,3846,3856],{},[73,3847,3848,3851,3854],{},[83,3849,3850],{},"ker",[76,3852,3853],{},"⁡",[83,3855,3196],{},[90,3857,3858],{"encoding":92},"\\ker \\phi",[57,3860,3862],{"className":3861,"ariaHidden":98},[97],[57,3863,3865,3868,3871,3874],{"className":3864},[102],[57,3866],{"className":3867,"style":1353},[106],[57,3869,3850],{"className":3870},[111],[57,3872],{"className":3873,"style":119},[118],[57,3875,3196],{"className":3876},[123,127]," 就是那些被映射成全是 0 的元素 ",[57,3879,3881,3894],{"className":3880},[60],[57,3882,3884],{"className":3883},[64],[66,3885,3886],{"xmlns":68},[70,3887,3888,3892],{},[73,3889,3890],{},[83,3891,483],{},[90,3893,483],{"encoding":92},[57,3895,3897],{"className":3896,"ariaHidden":98},[97],[57,3898,3900,3903],{"className":3899},[102],[57,3901],{"className":3902,"style":3686},[106],[57,3904,483],{"className":3905},[123,127],"。",[21,3908,3909,3910,4068,4069,4214],{},"这意味着 ",[57,3911,3913,3962],{"className":3912},[60],[57,3914,3916],{"className":3915},[64],[66,3917,3918],{"xmlns":68},[70,3919,3920,3959],{},[73,3921,3922,3924,3927,3930,3932,3935,3937,3948,3951,3957],{},[83,3923,483],{},[76,3925,3926],{},"≡",[278,3928,3929],{},"0",[118,3931],{},[118,3933],{"width":3934},"0.4444em",[76,3936,478],{"stretchy":477},[73,3938,3939,3942,3945],{},[83,3940,3941],{"mathvariant":302},"m",[83,3943,3944],{"mathvariant":302},"o",[83,3946,3947],{"mathvariant":302},"d",[118,3949],{"width":3950},"0.3333em",[80,3952,3953,3955],{},[83,3954,2354],{},[278,3956,280],{},[76,3958,513],{"stretchy":477},[90,3960,3961],{"encoding":92},"a \\equiv 0 \\pmod{I_1}",[57,3963,3965,3984,4001],{"className":3964,"ariaHidden":98},[97],[57,3966,3968,3972,3975,3978,3981],{"className":3967},[102],[57,3969],{"className":3970,"style":3971},[106],"height:0.4637em;",[57,3973,483],{"className":3974},[123,127],[57,3976],{"className":3977,"style":1281},[118],[57,3979,3926],{"className":3980},[1285],[57,3982],{"className":3983,"style":1281},[118],[57,3985,3987,3990,3993,3997],{"className":3986},[102],[57,3988],{"className":3989,"style":3167},[106],[57,3991,3929],{"className":3992},[123],[57,3994],{"className":3995},[118,3996],"allowbreak",[57,3998],{"className":3999,"style":4000},[118],"margin-right:0.4444em;",[57,4002,4004,4007,4010,4021,4025,4065],{"className":4003},[102],[57,4005],{"className":4006,"style":107},[106],[57,4008,478],{"className":4009},[529],[57,4011,4013],{"className":4012},[123],[57,4014,4016],{"className":4015},[123],[57,4017,4020],{"className":4018},[123,4019],"mathrm","mod",[57,4022],{"className":4023,"style":4024},[118],"margin-right:0.3333em;",[57,4026,4028,4031],{"className":4027},[123],[57,4029,2354],{"className":4030,"style":2387},[123,127],[57,4032,4034],{"className":4033},[132],[57,4035,4037,4057],{"className":4036},[136,137],[57,4038,4040,4054],{"className":4039},[141],[57,4041,4043],{"className":4042,"style":339},[145],[57,4044,4045,4048],{"style":2402},[57,4046],{"className":4047,"style":154},[153],[57,4049,4051],{"className":4050},[158,159,160,161],[57,4052,280],{"className":4053},[123,161],[57,4055,169],{"className":4056},[168],[57,4058,4060],{"className":4059},[141],[57,4061,4063],{"className":4062,"style":176},[145],[57,4064],{},[57,4066,513],{"className":4067},[679],"，且 ",[57,4070,4072,4114],{"className":4071},[60],[57,4073,4075],{"className":4074},[64],[66,4076,4077],{"xmlns":68},[70,4078,4079,4111],{},[73,4080,4081,4083,4085,4087,4089,4091,4093,4101,4103,4109],{},[83,4082,483],{},[76,4084,3926],{},[278,4086,3929],{},[118,4088],{},[118,4090],{"width":3934},[76,4092,478],{"stretchy":477},[73,4094,4095,4097,4099],{},[83,4096,3941],{"mathvariant":302},[83,4098,3944],{"mathvariant":302},[83,4100,3947],{"mathvariant":302},[118,4102],{"width":3950},[80,4104,4105,4107],{},[83,4106,2354],{},[278,4108,290],{},[76,4110,513],{"stretchy":477},[90,4112,4113],{"encoding":92},"a \\equiv 0 \\pmod{I_2}",[57,4115,4117,4135,4150],{"className":4116,"ariaHidden":98},[97],[57,4118,4120,4123,4126,4129,4132],{"className":4119},[102],[57,4121],{"className":4122,"style":3971},[106],[57,4124,483],{"className":4125},[123,127],[57,4127],{"className":4128,"style":1281},[118],[57,4130,3926],{"className":4131},[1285],[57,4133],{"className":4134,"style":1281},[118],[57,4136,4138,4141,4144,4147],{"className":4137},[102],[57,4139],{"className":4140,"style":3167},[106],[57,4142,3929],{"className":4143},[123],[57,4145],{"className":4146},[118,3996],[57,4148],{"className":4149,"style":4000},[118],[57,4151,4153,4156,4159,4168,4171,4211],{"className":4152},[102],[57,4154],{"className":4155,"style":107},[106],[57,4157,478],{"className":4158},[529],[57,4160,4162],{"className":4161},[123],[57,4163,4165],{"className":4164},[123],[57,4166,4020],{"className":4167},[123,4019],[57,4169],{"className":4170,"style":4024},[118],[57,4172,4174,4177],{"className":4173},[123],[57,4175,2354],{"className":4176,"style":2387},[123,127],[57,4178,4180],{"className":4179},[132],[57,4181,4183,4203],{"className":4182},[136,137],[57,4184,4186,4200],{"className":4185},[141],[57,4187,4189],{"className":4188,"style":339},[145],[57,4190,4191,4194],{"style":2402},[57,4192],{"className":4193,"style":154},[153],[57,4195,4197],{"className":4196},[158,159,160,161],[57,4198,290],{"className":4199},[123,161],[57,4201,169],{"className":4202},[168],[57,4204,4206],{"className":4205},[141],[57,4207,4209],{"className":4208,"style":176},[145],[57,4210],{},[57,4212,513],{"className":4213},[679],"……",[21,4216,4217,4218,4246,4247,3906],{},"也就是说 ",[57,4219,4221,4234],{"className":4220},[60],[57,4222,4224],{"className":4223},[64],[66,4225,4226],{"xmlns":68},[70,4227,4228,4232],{},[73,4229,4230],{},[83,4231,483],{},[90,4233,483],{"encoding":92},[57,4235,4237],{"className":4236,"ariaHidden":98},[97],[57,4238,4240,4243],{"className":4239},[102],[57,4241],{"className":4242,"style":3686},[106],[57,4244,483],{"className":4245},[123,127]," 必须同时属于 ",[57,4248,4250,4288],{"className":4249},[60],[57,4251,4253],{"className":4252},[64],[66,4254,4255],{"xmlns":68},[70,4256,4257,4285],{},[73,4258,4259,4265,4267,4273,4275,4277,4279],{},[80,4260,4261,4263],{},[83,4262,2354],{},[278,4264,280],{},[76,4266,283],{"separator":98},[80,4268,4269,4271],{},[83,4270,2354],{},[278,4272,290],{},[76,4274,283],{"separator":98},[76,4276,506],{},[76,4278,283],{"separator":98},[80,4280,4281,4283],{},[83,4282,2354],{},[83,4284,2206],{},[90,4286,4287],{"encoding":92},"I_1, I_2, \\dots, I_n",[57,4289,4291],{"className":4290,"ariaHidden":98},[97],[57,4292,4294,4297,4337,4340,4343,4383,4386,4389,4392,4395,4398,4401],{"className":4293},[102],[57,4295],{"className":4296,"style":320},[106],[57,4298,4300,4303],{"className":4299},[123],[57,4301,2354],{"className":4302,"style":2387},[123,127],[57,4304,4306],{"className":4305},[132],[57,4307,4309,4329],{"className":4308},[136,137],[57,4310,4312,4326],{"className":4311},[141],[57,4313,4315],{"className":4314,"style":339},[145],[57,4316,4317,4320],{"style":2402},[57,4318],{"className":4319,"style":154},[153],[57,4321,4323],{"className":4322},[158,159,160,161],[57,4324,280],{"className":4325},[123,161],[57,4327,169],{"className":4328},[168],[57,4330,4332],{"className":4331},[141],[57,4333,4335],{"className":4334,"style":176},[145],[57,4336],{},[57,4338,283],{"className":4339},[365],[57,4341],{"className":4342,"style":119},[118],[57,4344,4346,4349],{"className":4345},[123],[57,4347,2354],{"className":4348,"style":2387},[123,127],[57,4350,4352],{"className":4351},[132],[57,4353,4355,4375],{"className":4354},[136,137],[57,4356,4358,4372],{"className":4357},[141],[57,4359,4361],{"className":4360,"style":339},[145],[57,4362,4363,4366],{"style":2402},[57,4364],{"className":4365,"style":154},[153],[57,4367,4369],{"className":4368},[158,159,160,161],[57,4370,290],{"className":4371},[123,161],[57,4373,169],{"className":4374},[168],[57,4376,4378],{"className":4377},[141],[57,4379,4381],{"className":4380,"style":176},[145],[57,4382],{},[57,4384,283],{"className":4385},[365],[57,4387],{"className":4388,"style":119},[118],[57,4390,506],{"className":4391},[672],[57,4393],{"className":4394,"style":119},[118],[57,4396,283],{"className":4397},[365],[57,4399],{"className":4400,"style":119},[118],[57,4402,4404,4407],{"className":4403},[123],[57,4405,2354],{"className":4406,"style":2387},[123,127],[57,4408,4410],{"className":4409},[132],[57,4411,4413,4433],{"className":4412},[136,137],[57,4414,4416,4430],{"className":4415},[141],[57,4417,4419],{"className":4418,"style":2295},[145],[57,4420,4421,4424],{"style":2402},[57,4422],{"className":4423,"style":154},[153],[57,4425,4427],{"className":4426},[158,159,160,161],[57,4428,2206],{"className":4429},[123,127,161],[57,4431,169],{"className":4432},[168],[57,4434,4436],{"className":4435},[141],[57,4437,4439],{"className":4438,"style":176},[145],[57,4440],{},[21,4442,4443,4444,3906],{},"所以，",[57,4445,4447,4505],{"className":4446},[60],[57,4448,4450],{"className":4449},[64],[66,4451,4452],{"xmlns":68},[70,4453,4454,4502],{},[73,4455,4456,4458,4460,4462,4464,4470,4473,4479,4481,4484,4486,4492,4494,4496],{},[83,4457,3850],{},[76,4459,3853],{},[83,4461,3196],{},[76,4463,1030],{},[80,4465,4466,4468],{},[83,4467,2354],{},[278,4469,280],{},[76,4471,4472],{},"∩",[80,4474,4475,4477],{},[83,4476,2354],{},[278,4478,290],{},[76,4480,4472],{},[76,4482,4483],{},"⋯",[76,4485,4472],{},[80,4487,4488,4490],{},[83,4489,2354],{},[83,4491,2206],{},[76,4493,1030],{},[76,4495,2540],{},[80,4497,4498,4500],{},[83,4499,2354],{},[83,4501,2545],{},[90,4503,4504],{"encoding":92},"\\ker \\phi = I_1 \\cap I_2 \\cap \\dots \\cap I_n = \\bigcap I_k",[57,4506,4508,4532,4587,4642,4661,4716],{"className":4507,"ariaHidden":98},[97],[57,4509,4511,4514,4517,4520,4523,4526,4529],{"className":4510},[102],[57,4512],{"className":4513,"style":1353},[106],[57,4515,3850],{"className":4516},[111],[57,4518],{"className":4519,"style":119},[118],[57,4521,3196],{"className":4522},[123,127],[57,4524],{"className":4525,"style":1281},[118],[57,4527,1030],{"className":4528},[1285],[57,4530],{"className":4531,"style":1281},[118],[57,4533,4535,4538,4578,4581,4584],{"className":4534},[102],[57,4536],{"className":4537,"style":219},[106],[57,4539,4541,4544],{"className":4540},[123],[57,4542,2354],{"className":4543,"style":2387},[123,127],[57,4545,4547],{"className":4546},[132],[57,4548,4550,4570],{"className":4549},[136,137],[57,4551,4553,4567],{"className":4552},[141],[57,4554,4556],{"className":4555,"style":339},[145],[57,4557,4558,4561],{"style":2402},[57,4559],{"className":4560,"style":154},[153],[57,4562,4564],{"className":4563},[158,159,160,161],[57,4565,280],{"className":4566},[123,161],[57,4568,169],{"className":4569},[168],[57,4571,4573],{"className":4572},[141],[57,4574,4576],{"className":4575,"style":176},[145],[57,4577],{},[57,4579],{"className":4580,"style":1172},[118],[57,4582,4472],{"className":4583},[1176],[57,4585],{"className":4586,"style":1172},[118],[57,4588,4590,4593,4633,4636,4639],{"className":4589},[102],[57,4591],{"className":4592,"style":219},[106],[57,4594,4596,4599],{"className":4595},[123],[57,4597,2354],{"className":4598,"style":2387},[123,127],[57,4600,4602],{"className":4601},[132],[57,4603,4605,4625],{"className":4604},[136,137],[57,4606,4608,4622],{"className":4607},[141],[57,4609,4611],{"className":4610,"style":339},[145],[57,4612,4613,4616],{"style":2402},[57,4614],{"className":4615,"style":154},[153],[57,4617,4619],{"className":4618},[158,159,160,161],[57,4620,290],{"className":4621},[123,161],[57,4623,169],{"className":4624},[168],[57,4626,4628],{"className":4627},[141],[57,4629,4631],{"className":4630,"style":176},[145],[57,4632],{},[57,4634],{"className":4635,"style":1172},[118],[57,4637,4472],{"className":4638},[1176],[57,4640],{"className":4641,"style":1172},[118],[57,4643,4645,4649,4652,4655,4658],{"className":4644},[102],[57,4646],{"className":4647,"style":4648},[106],"height:0.5556em;",[57,4650,4483],{"className":4651},[672],[57,4653],{"className":4654,"style":1172},[118],[57,4656,4472],{"className":4657},[1176],[57,4659],{"className":4660,"style":1172},[118],[57,4662,4664,4667,4707,4710,4713],{"className":4663},[102],[57,4665],{"className":4666,"style":219},[106],[57,4668,4670,4673],{"className":4669},[123],[57,4671,2354],{"className":4672,"style":2387},[123,127],[57,4674,4676],{"className":4675},[132],[57,4677,4679,4699],{"className":4678},[136,137],[57,4680,4682,4696],{"className":4681},[141],[57,4683,4685],{"className":4684,"style":2295},[145],[57,4686,4687,4690],{"style":2402},[57,4688],{"className":4689,"style":154},[153],[57,4691,4693],{"className":4692},[158,159,160,161],[57,4694,2206],{"className":4695},[123,127,161],[57,4697,169],{"className":4698},[168],[57,4700,4702],{"className":4701},[141],[57,4703,4705],{"className":4704,"style":176},[145],[57,4706],{},[57,4708],{"className":4709,"style":1281},[118],[57,4711,1030],{"className":4712},[1285],[57,4714],{"className":4715,"style":1281},[118],[57,4717,4719,4722,4725,4728],{"className":4718},[102],[57,4720],{"className":4721,"style":107},[106],[57,4723,2540],{"className":4724,"style":114},[111,112,113],[57,4726],{"className":4727,"style":119},[118],[57,4729,4731,4734],{"className":4730},[123],[57,4732,2354],{"className":4733,"style":2387},[123,127],[57,4735,4737],{"className":4736},[132],[57,4738,4740,4760],{"className":4739},[136,137],[57,4741,4743,4757],{"className":4742},[141],[57,4744,4746],{"className":4745,"style":2695},[145],[57,4747,4748,4751],{"style":2402},[57,4749],{"className":4750,"style":154},[153],[57,4752,4754],{"className":4753},[158,159,160,161],[57,4755,2545],{"className":4756,"style":2643},[123,127,161],[57,4758,169],{"className":4759},[168],[57,4761,4763],{"className":4762},[141],[57,4764,4766],{"className":4765,"style":176},[145],[57,4767],{},[14,4769,4770,4771,55,4774,4858,4859,4862],{},"根据",[24,4772,4773],{},"同态基本定理",[57,4775,4777,4806],{"className":4776},[60],[57,4778,4780],{"className":4779},[64],[66,4781,4782],{"xmlns":68},[70,4783,4784,4803],{},[73,4785,4786,4788,4790,4792,4794,4796,4798,4801],{},[83,4787,85],{},[83,4789,2532],{"mathvariant":302},[83,4791,3850],{},[76,4793,3853],{},[83,4795,3196],{},[76,4797,2562],{},[508,4799,4800],{},"Im ",[83,4802,3196],{},[90,4804,4805],{"encoding":92},"R \u002F \\ker \\phi \\cong \\text{Im } \\phi",[57,4807,4809,4842],{"className":4808,"ariaHidden":98},[97],[57,4810,4812,4815,4818,4821,4824,4827,4830,4833,4836,4839],{"className":4811},[102],[57,4813],{"className":4814,"style":107},[106],[57,4816,85],{"className":4817,"style":128},[123,127],[57,4819,2532],{"className":4820},[123],[57,4822],{"className":4823,"style":119},[118],[57,4825,3850],{"className":4826},[111],[57,4828],{"className":4829,"style":119},[118],[57,4831,3196],{"className":4832},[123,127],[57,4834],{"className":4835,"style":1281},[118],[57,4837,2562],{"className":4838},[1285],[57,4840],{"className":4841,"style":1281},[118],[57,4843,4845,4848,4855],{"className":4844},[102],[57,4846],{"className":4847,"style":1353},[106],[57,4849,4852],{"className":4850},[123,4851],"text",[57,4853,4800],{"className":4854},[123],[57,4856,3196],{"className":4857},[123,127],"。\n现在左边有了，我们只差最后一步：",[24,4860,4861],{},"证明右边是满射 (Surjective)","。\n也就是要证明：对于右边任意一个列表，我们都能在左边找到对应的原像。",[35,4864],{},[46,4866,4868],{"id":4867},"证明第二步证明满射最难的归纳法部分","证明第二步：证明满射（最难的归纳法部分）",[14,4870,4871,4872,4955,4956,4984,4985,5055],{},"我们要证明满射，本质上是要构造一组“开关”。\n比如我想构造列表 ",[57,4873,4875,4907],{"className":4874},[60],[57,4876,4878],{"className":4877},[64],[66,4879,4880],{"xmlns":68},[70,4881,4882,4904],{},[73,4883,4884,4886,4888,4890,4892,4894,4896,4898,4900,4902],{},[76,4885,478],{"stretchy":477},[278,4887,280],{},[76,4889,283],{"separator":98},[278,4891,3929],{},[76,4893,283],{"separator":98},[278,4895,3929],{},[76,4897,283],{"separator":98},[76,4899,506],{},[508,4901,510],{},[76,4903,513],{"stretchy":477},[90,4905,4906],{"encoding":92},"(1, 0, 0, \\dots)",[57,4908,4910],{"className":4909,"ariaHidden":98},[97],[57,4911,4913,4916,4919,4922,4925,4928,4931,4934,4937,4940,4943,4946,4949,4952],{"className":4912},[102],[57,4914],{"className":4915,"style":107},[106],[57,4917,478],{"className":4918},[529],[57,4920,280],{"className":4921},[123],[57,4923,283],{"className":4924},[365],[57,4926],{"className":4927,"style":119},[118],[57,4929,3929],{"className":4930},[123],[57,4932,283],{"className":4933},[365],[57,4935],{"className":4936,"style":119},[118],[57,4938,3929],{"className":4939},[123],[57,4941,283],{"className":4942},[365],[57,4944],{"className":4945,"style":119},[118],[57,4947,506],{"className":4948},[672],[57,4950],{"className":4951,"style":119},[118],[57,4953,513],{"className":4954},[679],"，我需要找到一个 ",[57,4957,4959,4972],{"className":4958},[60],[57,4960,4962],{"className":4961},[64],[66,4963,4964],{"xmlns":68},[70,4965,4966,4970],{},[73,4967,4968],{},[83,4969,3000],{},[90,4971,3000],{"encoding":92},[57,4973,4975],{"className":4974,"ariaHidden":98},[97],[57,4976,4978,4981],{"className":4977},[102],[57,4979],{"className":4980,"style":3686},[106],[57,4982,3000],{"className":4983},[123,127],"，它模 ",[57,4986,4988,5006],{"className":4987},[60],[57,4989,4991],{"className":4990},[64],[66,4992,4993],{"xmlns":68},[70,4994,4995,5003],{},[73,4996,4997],{},[80,4998,4999,5001],{},[83,5000,2354],{},[278,5002,280],{},[90,5004,5005],{"encoding":92},"I_1",[57,5007,5009],{"className":5008,"ariaHidden":98},[97],[57,5010,5012,5015],{"className":5011},[102],[57,5013],{"className":5014,"style":219},[106],[57,5016,5018,5021],{"className":5017},[123],[57,5019,2354],{"className":5020,"style":2387},[123,127],[57,5022,5024],{"className":5023},[132],[57,5025,5027,5047],{"className":5026},[136,137],[57,5028,5030,5044],{"className":5029},[141],[57,5031,5033],{"className":5032,"style":339},[145],[57,5034,5035,5038],{"style":2402},[57,5036],{"className":5037,"style":154},[153],[57,5039,5041],{"className":5040},[158,159,160,161],[57,5042,280],{"className":5043},[123,161],[57,5045,169],{"className":5046},[168],[57,5048,5050],{"className":5049},[141],[57,5051,5053],{"className":5052,"style":176},[145],[57,5054],{}," 是 1，模其他都是 0。",[14,5057,5058],{},[24,5059,5060],{},"课件里的构造逻辑：",[18,5062,5063,6430,8709,11252],{},[21,5064,5065,5141,5142,5386],{},[24,5066,5067,5068],{},"定义 ",[57,5069,5071,5090],{"className":5070},[60],[57,5072,5074],{"className":5073},[64],[66,5075,5076],{"xmlns":68},[70,5077,5078,5087],{},[73,5079,5080],{},[80,5081,5082,5085],{},[83,5083,5084],{},"M",[83,5086,88],{},[90,5088,5089],{"encoding":92},"M_i",[57,5091,5093],{"className":5092,"ariaHidden":98},[97],[57,5094,5096,5099],{"className":5095},[102],[57,5097],{"className":5098,"style":219},[106],[57,5100,5102,5106],{"className":5101},[123],[57,5103,5084],{"className":5104,"style":5105},[123,127],"margin-right:0.109em;",[57,5107,5109],{"className":5108},[132],[57,5110,5112,5133],{"className":5111},[136,137],[57,5113,5115,5130],{"className":5114},[141],[57,5116,5118],{"className":5117,"style":146},[145],[57,5119,5121,5124],{"style":5120},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[57,5122],{"className":5123,"style":154},[153],[57,5125,5127],{"className":5126},[158,159,160,161],[57,5128,88],{"className":5129},[123,127,161],[57,5131,169],{"className":5132},[168],[57,5134,5136],{"className":5135},[141],[57,5137,5139],{"className":5138,"style":176},[145],[57,5140],{},"：\n",[57,5143,5145,5185],{"className":5144},[60],[57,5146,5148],{"className":5147},[64],[66,5149,5150],{"xmlns":68},[70,5151,5152,5182],{},[73,5153,5154,5160,5162,5176],{},[80,5155,5156,5158],{},[83,5157,5084],{},[83,5159,88],{},[76,5161,1030],{},[80,5163,5164,5166],{},[76,5165,78],{},[73,5167,5168,5171,5174],{},[83,5169,5170],{},"j",[76,5172,5173],{"mathvariant":302},"≠",[83,5175,88],{},[80,5177,5178,5180],{},[83,5179,2354],{},[83,5181,5170],{},[90,5183,5184],{"encoding":92},"M_i = \\prod_{j \\neq i} I_j",[57,5186,5188,5243],{"className":5187,"ariaHidden":98},[97],[57,5189,5191,5194,5234,5237,5240],{"className":5190},[102],[57,5192],{"className":5193,"style":219},[106],[57,5195,5197,5200],{"className":5196},[123],[57,5198,5084],{"className":5199,"style":5105},[123,127],[57,5201,5203],{"className":5202},[132],[57,5204,5206,5226],{"className":5205},[136,137],[57,5207,5209,5223],{"className":5208},[141],[57,5210,5212],{"className":5211,"style":146},[145],[57,5213,5214,5217],{"style":5120},[57,5215],{"className":5216,"style":154},[153],[57,5218,5220],{"className":5219},[158,159,160,161],[57,5221,88],{"className":5222},[123,127,161],[57,5224,169],{"className":5225},[168],[57,5227,5229],{"className":5228},[141],[57,5230,5232],{"className":5231,"style":176},[145],[57,5233],{},[57,5235],{"className":5236,"style":1281},[118],[57,5238,1030],{"className":5239},[1285],[57,5241],{"className":5242,"style":1281},[118],[57,5244,5246,5250,5342,5345],{"className":5245},[102],[57,5247],{"className":5248,"style":5249},[106],"height:1.1858em;vertical-align:-0.4358em;",[57,5251,5253,5256],{"className":5252},[111],[57,5254,78],{"className":5255,"style":114},[111,112,113],[57,5257,5259],{"className":5258},[132],[57,5260,5262,5333],{"className":5261},[136,137],[57,5263,5265,5330],{"className":5264},[141],[57,5266,5269],{"className":5267,"style":5268},[145],"height:0.1864em;",[57,5270,5271,5274],{"style":2630},[57,5272],{"className":5273,"style":154},[153],[57,5275,5277],{"className":5276},[158,159,160,161],[57,5278,5280,5284,5327],{"className":5279},[123,161],[57,5281,5170],{"className":5282,"style":5283},[123,127,161],"margin-right:0.0572em;",[57,5285,5287,5320,5324],{"className":5286},[1285,161],[57,5288,5290],{"className":5289},[1285,161],[57,5291,5294],{"className":5292},[123,5293,161],"vbox",[57,5295,5298],{"className":5296},[5297,161],"thinbox",[57,5299,5302,5305,5316],{"className":5300},[5301,161],"rlap",[57,5303],{"className":5304,"style":1353},[106],[57,5306,5309],{"className":5307},[5308],"inner",[57,5310,5312],{"className":5311},[123,161],[57,5313,5315],{"className":5314},[1285,161],"",[57,5317],{"className":5318},[5319],"fix",[57,5321],{"className":5322},[118,5323,161],"nobreak",[57,5325,1030],{"className":5326},[1285,161],[57,5328,88],{"className":5329},[123,127,161],[57,5331,169],{"className":5332},[168],[57,5334,5336],{"className":5335},[141],[57,5337,5340],{"className":5338,"style":5339},[145],"height:0.4358em;",[57,5341],{},[57,5343],{"className":5344,"style":119},[118],[57,5346,5348,5351],{"className":5347},[123],[57,5349,2354],{"className":5350,"style":2387},[123,127],[57,5352,5354],{"className":5353},[132],[57,5355,5357,5377],{"className":5356},[136,137],[57,5358,5360,5374],{"className":5359},[141],[57,5361,5363],{"className":5362,"style":146},[145],[57,5364,5365,5368],{"style":2402},[57,5366],{"className":5367,"style":154},[153],[57,5369,5371],{"className":5370},[158,159,160,161],[57,5372,5170],{"className":5373,"style":5283},[123,127,161],[57,5375,169],{"className":5376},[168],[57,5378,5380],{"className":5379},[141],[57,5381,5384],{"className":5382,"style":5383},[145],"height:0.2861em;",[57,5385],{},[184,5387,5388,5724,6135],{},[21,5389,5390,55,5392,5462,5463,5653,5654,5723],{},[24,5391,3630],{},[57,5393,5395,5413],{"className":5394},[60],[57,5396,5398],{"className":5397},[64],[66,5399,5400],{"xmlns":68},[70,5401,5402,5410],{},[73,5403,5404],{},[80,5405,5406,5408],{},[83,5407,5084],{},[278,5409,280],{},[90,5411,5412],{"encoding":92},"M_1",[57,5414,5416],{"className":5415,"ariaHidden":98},[97],[57,5417,5419,5422],{"className":5418},[102],[57,5420],{"className":5421,"style":219},[106],[57,5423,5425,5428],{"className":5424},[123],[57,5426,5084],{"className":5427,"style":5105},[123,127],[57,5429,5431],{"className":5430},[132],[57,5432,5434,5454],{"className":5433},[136,137],[57,5435,5437,5451],{"className":5436},[141],[57,5438,5440],{"className":5439,"style":339},[145],[57,5441,5442,5445],{"style":5120},[57,5443],{"className":5444,"style":154},[153],[57,5446,5448],{"className":5447},[158,159,160,161],[57,5449,280],{"className":5450},[123,161],[57,5452,169],{"className":5453},[168],[57,5455,5457],{"className":5456},[141],[57,5458,5460],{"className":5459,"style":176},[145],[57,5461],{}," 是 ",[57,5464,5466,5500],{"className":5465},[60],[57,5467,5469],{"className":5468},[64],[66,5470,5471],{"xmlns":68},[70,5472,5473,5497],{},[73,5474,5475,5481,5483,5489,5491],{},[80,5476,5477,5479],{},[83,5478,2354],{},[278,5480,290],{},[76,5482,1533],{},[80,5484,5485,5487],{},[83,5486,2354],{},[278,5488,299],{},[76,5490,4483],{},[80,5492,5493,5495],{},[83,5494,2354],{},[83,5496,2206],{},[90,5498,5499],{"encoding":92},"I_2 \\cdot I_3 \\cdots I_n",[57,5501,5503,5558],{"className":5502,"ariaHidden":98},[97],[57,5504,5506,5509,5549,5552,5555],{"className":5505},[102],[57,5507],{"className":5508,"style":219},[106],[57,5510,5512,5515],{"className":5511},[123],[57,5513,2354],{"className":5514,"style":2387},[123,127],[57,5516,5518],{"className":5517},[132],[57,5519,5521,5541],{"className":5520},[136,137],[57,5522,5524,5538],{"className":5523},[141],[57,5525,5527],{"className":5526,"style":339},[145],[57,5528,5529,5532],{"style":2402},[57,5530],{"className":5531,"style":154},[153],[57,5533,5535],{"className":5534},[158,159,160,161],[57,5536,290],{"className":5537},[123,161],[57,5539,169],{"className":5540},[168],[57,5542,5544],{"className":5543},[141],[57,5545,5547],{"className":5546,"style":176},[145],[57,5548],{},[57,5550],{"className":5551,"style":1172},[118],[57,5553,1533],{"className":5554},[1176],[57,5556],{"className":5557,"style":1172},[118],[57,5559,5561,5564,5604,5607,5610,5613],{"className":5560},[102],[57,5562],{"className":5563,"style":219},[106],[57,5565,5567,5570],{"className":5566},[123],[57,5568,2354],{"className":5569,"style":2387},[123,127],[57,5571,5573],{"className":5572},[132],[57,5574,5576,5596],{"className":5575},[136,137],[57,5577,5579,5593],{"className":5578},[141],[57,5580,5582],{"className":5581,"style":339},[145],[57,5583,5584,5587],{"style":2402},[57,5585],{"className":5586,"style":154},[153],[57,5588,5590],{"className":5589},[158,159,160,161],[57,5591,299],{"className":5592},[123,161],[57,5594,169],{"className":5595},[168],[57,5597,5599],{"className":5598},[141],[57,5600,5602],{"className":5601,"style":176},[145],[57,5603],{},[57,5605],{"className":5606,"style":119},[118],[57,5608,4483],{"className":5609},[672],[57,5611],{"className":5612,"style":119},[118],[57,5614,5616,5619],{"className":5615},[123],[57,5617,2354],{"className":5618,"style":2387},[123,127],[57,5620,5622],{"className":5621},[132],[57,5623,5625,5645],{"className":5624},[136,137],[57,5626,5628,5642],{"className":5627},[141],[57,5629,5631],{"className":5630,"style":2295},[145],[57,5632,5633,5636],{"style":2402},[57,5634],{"className":5635,"style":154},[153],[57,5637,5639],{"className":5638},[158,159,160,161],[57,5640,2206],{"className":5641},[123,127,161],[57,5643,169],{"className":5644},[168],[57,5646,5648],{"className":5647},[141],[57,5649,5651],{"className":5650,"style":176},[145],[57,5652],{},"（除了 ",[57,5655,5657,5674],{"className":5656},[60],[57,5658,5660],{"className":5659},[64],[66,5661,5662],{"xmlns":68},[70,5663,5664,5672],{},[73,5665,5666],{},[80,5667,5668,5670],{},[83,5669,2354],{},[278,5671,280],{},[90,5673,5005],{"encoding":92},[57,5675,5677],{"className":5676,"ariaHidden":98},[97],[57,5678,5680,5683],{"className":5679},[102],[57,5681],{"className":5682,"style":219},[106],[57,5684,5686,5689],{"className":5685},[123],[57,5687,2354],{"className":5688,"style":2387},[123,127],[57,5690,5692],{"className":5691},[132],[57,5693,5695,5715],{"className":5694},[136,137],[57,5696,5698,5712],{"className":5697},[141],[57,5699,5701],{"className":5700,"style":339},[145],[57,5702,5703,5706],{"style":2402},[57,5704],{"className":5705,"style":154},[153],[57,5707,5709],{"className":5708},[158,159,160,161],[57,5710,280],{"className":5711},[123,161],[57,5713,169],{"className":5714},[168],[57,5716,5718],{"className":5717},[141],[57,5719,5721],{"className":5720,"style":176},[145],[57,5722],{}," 以外所有理想的乘积）。",[21,5725,5726,5729,5730,5799,5800,5932,5933,6002,6003,6134],{},[24,5727,5728],{},"关键性质","：因为 ",[57,5731,5733,5750],{"className":5732},[60],[57,5734,5736],{"className":5735},[64],[66,5737,5738],{"xmlns":68},[70,5739,5740,5748],{},[73,5741,5742],{},[80,5743,5744,5746],{},[83,5745,5084],{},[278,5747,280],{},[90,5749,5412],{"encoding":92},[57,5751,5753],{"className":5752,"ariaHidden":98},[97],[57,5754,5756,5759],{"className":5755},[102],[57,5757],{"className":5758,"style":219},[106],[57,5760,5762,5765],{"className":5761},[123],[57,5763,5084],{"className":5764,"style":5105},[123,127],[57,5766,5768],{"className":5767},[132],[57,5769,5771,5791],{"className":5770},[136,137],[57,5772,5774,5788],{"className":5773},[141],[57,5775,5777],{"className":5776,"style":339},[145],[57,5778,5779,5782],{"style":5120},[57,5780],{"className":5781,"style":154},[153],[57,5783,5785],{"className":5784},[158,159,160,161],[57,5786,280],{"className":5787},[123,161],[57,5789,169],{"className":5790},[168],[57,5792,5794],{"className":5793},[141],[57,5795,5797],{"className":5796,"style":176},[145],[57,5798],{}," 包含 ",[57,5801,5803,5831],{"className":5802},[60],[57,5804,5806],{"className":5805},[64],[66,5807,5808],{"xmlns":68},[70,5809,5810,5828],{},[73,5811,5812,5818,5820,5826],{},[80,5813,5814,5816],{},[83,5815,2354],{},[278,5817,290],{},[76,5819,283],{"separator":98},[80,5821,5822,5824],{},[83,5823,2354],{},[278,5825,299],{},[76,5827,506],{},[90,5829,5830],{"encoding":92},"I_2, I_3 \\dots",[57,5832,5834],{"className":5833,"ariaHidden":98},[97],[57,5835,5837,5840,5880,5883,5886,5926,5929],{"className":5836},[102],[57,5838],{"className":5839,"style":320},[106],[57,5841,5843,5846],{"className":5842},[123],[57,5844,2354],{"className":5845,"style":2387},[123,127],[57,5847,5849],{"className":5848},[132],[57,5850,5852,5872],{"className":5851},[136,137],[57,5853,5855,5869],{"className":5854},[141],[57,5856,5858],{"className":5857,"style":339},[145],[57,5859,5860,5863],{"style":2402},[57,5861],{"className":5862,"style":154},[153],[57,5864,5866],{"className":5865},[158,159,160,161],[57,5867,290],{"className":5868},[123,161],[57,5870,169],{"className":5871},[168],[57,5873,5875],{"className":5874},[141],[57,5876,5878],{"className":5877,"style":176},[145],[57,5879],{},[57,5881,283],{"className":5882},[365],[57,5884],{"className":5885,"style":119},[118],[57,5887,5889,5892],{"className":5888},[123],[57,5890,2354],{"className":5891,"style":2387},[123,127],[57,5893,5895],{"className":5894},[132],[57,5896,5898,5918],{"className":5897},[136,137],[57,5899,5901,5915],{"className":5900},[141],[57,5902,5904],{"className":5903,"style":339},[145],[57,5905,5906,5909],{"style":2402},[57,5907],{"className":5908,"style":154},[153],[57,5910,5912],{"className":5911},[158,159,160,161],[57,5913,299],{"className":5914},[123,161],[57,5916,169],{"className":5917},[168],[57,5919,5921],{"className":5920},[141],[57,5922,5924],{"className":5923,"style":176},[145],[57,5925],{},[57,5927],{"className":5928,"style":119},[118],[57,5930,506],{"className":5931},[672]," 的因子，所以 ",[57,5934,5936,5953],{"className":5935},[60],[57,5937,5939],{"className":5938},[64],[66,5940,5941],{"xmlns":68},[70,5942,5943,5951],{},[73,5944,5945],{},[80,5946,5947,5949],{},[83,5948,5084],{},[278,5950,280],{},[90,5952,5412],{"encoding":92},[57,5954,5956],{"className":5955,"ariaHidden":98},[97],[57,5957,5959,5962],{"className":5958},[102],[57,5960],{"className":5961,"style":219},[106],[57,5963,5965,5968],{"className":5964},[123],[57,5966,5084],{"className":5967,"style":5105},[123,127],[57,5969,5971],{"className":5970},[132],[57,5972,5974,5994],{"className":5973},[136,137],[57,5975,5977,5991],{"className":5976},[141],[57,5978,5980],{"className":5979,"style":339},[145],[57,5981,5982,5985],{"style":5120},[57,5983],{"className":5984,"style":154},[153],[57,5986,5988],{"className":5987},[158,159,160,161],[57,5989,280],{"className":5990},[123,161],[57,5992,169],{"className":5993},[168],[57,5995,5997],{"className":5996},[141],[57,5998,6000],{"className":5999,"style":176},[145],[57,6001],{}," 落在 ",[57,6004,6006,6033],{"className":6005},[60],[57,6007,6009],{"className":6008},[64],[66,6010,6011],{"xmlns":68},[70,6012,6013,6031],{},[73,6014,6015,6021,6023,6029],{},[80,6016,6017,6019],{},[83,6018,2354],{},[278,6020,290],{},[76,6022,283],{"separator":98},[80,6024,6025,6027],{},[83,6026,2354],{},[278,6028,299],{},[76,6030,506],{},[90,6032,5830],{"encoding":92},[57,6034,6036],{"className":6035,"ariaHidden":98},[97],[57,6037,6039,6042,6082,6085,6088,6128,6131],{"className":6038},[102],[57,6040],{"className":6041,"style":320},[106],[57,6043,6045,6048],{"className":6044},[123],[57,6046,2354],{"className":6047,"style":2387},[123,127],[57,6049,6051],{"className":6050},[132],[57,6052,6054,6074],{"className":6053},[136,137],[57,6055,6057,6071],{"className":6056},[141],[57,6058,6060],{"className":6059,"style":339},[145],[57,6061,6062,6065],{"style":2402},[57,6063],{"className":6064,"style":154},[153],[57,6066,6068],{"className":6067},[158,159,160,161],[57,6069,290],{"className":6070},[123,161],[57,6072,169],{"className":6073},[168],[57,6075,6077],{"className":6076},[141],[57,6078,6080],{"className":6079,"style":176},[145],[57,6081],{},[57,6083,283],{"className":6084},[365],[57,6086],{"className":6087,"style":119},[118],[57,6089,6091,6094],{"className":6090},[123],[57,6092,2354],{"className":6093,"style":2387},[123,127],[57,6095,6097],{"className":6096},[132],[57,6098,6100,6120],{"className":6099},[136,137],[57,6101,6103,6117],{"className":6102},[141],[57,6104,6106],{"className":6105,"style":339},[145],[57,6107,6108,6111],{"style":2402},[57,6109],{"className":6110,"style":154},[153],[57,6112,6114],{"className":6113},[158,159,160,161],[57,6115,299],{"className":6116},[123,161],[57,6118,169],{"className":6119},[168],[57,6121,6123],{"className":6122},[141],[57,6124,6126],{"className":6125,"style":176},[145],[57,6127],{},[57,6129],{"className":6130,"style":119},[118],[57,6132,506],{"className":6133},[672]," 里面。",[21,6136,6137,6138,6222,6223,6358,6359,6429],{},"也就是说：对于任何 ",[57,6139,6141,6159],{"className":6140},[60],[57,6142,6144],{"className":6143},[64],[66,6145,6146],{"xmlns":68},[70,6147,6148,6156],{},[73,6149,6150,6152,6154],{},[83,6151,5170],{},[76,6153,5173],{"mathvariant":302},[278,6155,280],{},[90,6157,6158],{"encoding":92},"j \\neq 1",[57,6160,6162,6213],{"className":6161,"ariaHidden":98},[97],[57,6163,6165,6168,6171,6174,6210],{"className":6164},[102],[57,6166],{"className":6167,"style":1353},[106],[57,6169,5170],{"className":6170,"style":5283},[123,127],[57,6172],{"className":6173,"style":1281},[118],[57,6175,6177,6204,6207],{"className":6176},[1285],[57,6178,6180],{"className":6179},[1285],[57,6181,6183],{"className":6182},[123,5293],[57,6184,6186],{"className":6185},[5297],[57,6187,6189,6192,6201],{"className":6188},[5301],[57,6190],{"className":6191,"style":1353},[106],[57,6193,6195],{"className":6194},[5308],[57,6196,6198],{"className":6197},[123],[57,6199,5315],{"className":6200},[1285],[57,6202],{"className":6203},[5319],[57,6205],{"className":6206},[118,5323],[57,6208,1030],{"className":6209},[1285],[57,6211],{"className":6212,"style":1281},[118],[57,6214,6216,6219],{"className":6215},[102],[57,6217],{"className":6218,"style":3167},[106],[57,6220,280],{"className":6221},[123],"，都有 ",[57,6224,6226,6253],{"className":6225},[60],[57,6227,6229],{"className":6228},[64],[66,6230,6231],{"xmlns":68},[70,6232,6233,6250],{},[73,6234,6235,6241,6244],{},[80,6236,6237,6239],{},[83,6238,5084],{},[278,6240,280],{},[76,6242,6243],{},"⊆",[80,6245,6246,6248],{},[83,6247,2354],{},[83,6249,5170],{},[90,6251,6252],{"encoding":92},"M_1 \\subseteq I_j",[57,6254,6256,6311],{"className":6255,"ariaHidden":98},[97],[57,6257,6259,6262,6302,6305,6308],{"className":6258},[102],[57,6260],{"className":6261,"style":219},[106],[57,6263,6265,6268],{"className":6264},[123],[57,6266,5084],{"className":6267,"style":5105},[123,127],[57,6269,6271],{"className":6270},[132],[57,6272,6274,6294],{"className":6273},[136,137],[57,6275,6277,6291],{"className":6276},[141],[57,6278,6280],{"className":6279,"style":339},[145],[57,6281,6282,6285],{"style":5120},[57,6283],{"className":6284,"style":154},[153],[57,6286,6288],{"className":6287},[158,159,160,161],[57,6289,280],{"className":6290},[123,161],[57,6292,169],{"className":6293},[168],[57,6295,6297],{"className":6296},[141],[57,6298,6300],{"className":6299,"style":176},[145],[57,6301],{},[57,6303],{"className":6304,"style":1281},[118],[57,6306,6243],{"className":6307},[1285],[57,6309],{"className":6310,"style":1281},[118],[57,6312,6314,6318],{"className":6313},[102],[57,6315],{"className":6316,"style":6317},[106],"height:0.9694em;vertical-align:-0.2861em;",[57,6319,6321,6324],{"className":6320},[123],[57,6322,2354],{"className":6323,"style":2387},[123,127],[57,6325,6327],{"className":6326},[132],[57,6328,6330,6350],{"className":6329},[136,137],[57,6331,6333,6347],{"className":6332},[141],[57,6334,6336],{"className":6335,"style":146},[145],[57,6337,6338,6341],{"style":2402},[57,6339],{"className":6340,"style":154},[153],[57,6342,6344],{"className":6343},[158,159,160,161],[57,6345,5170],{"className":6346,"style":5283},[123,127,161],[57,6348,169],{"className":6349},[168],[57,6351,6353],{"className":6352},[141],[57,6354,6356],{"className":6355,"style":5383},[145],[57,6357],{},"（即模 ",[57,6360,6362,6380],{"className":6361},[60],[57,6363,6365],{"className":6364},[64],[66,6366,6367],{"xmlns":68},[70,6368,6369,6377],{},[73,6370,6371],{},[80,6372,6373,6375],{},[83,6374,2354],{},[83,6376,5170],{},[90,6378,6379],{"encoding":92},"I_j",[57,6381,6383],{"className":6382,"ariaHidden":98},[97],[57,6384,6386,6389],{"className":6385},[102],[57,6387],{"className":6388,"style":6317},[106],[57,6390,6392,6395],{"className":6391},[123],[57,6393,2354],{"className":6394,"style":2387},[123,127],[57,6396,6398],{"className":6397},[132],[57,6399,6401,6421],{"className":6400},[136,137],[57,6402,6404,6418],{"className":6403},[141],[57,6405,6407],{"className":6406,"style":146},[145],[57,6408,6409,6412],{"style":2402},[57,6410],{"className":6411,"style":154},[153],[57,6413,6415],{"className":6414},[158,159,160,161],[57,6416,5170],{"className":6417,"style":5283},[123,127,161],[57,6419,169],{"className":6420},[168],[57,6422,6424],{"className":6423},[141],[57,6425,6427],{"className":6426,"style":5383},[145],[57,6428],{}," 为 0）。",[21,6431,6432,55,6435,6675,6676,3906,6678,6681,6684],{},[24,6433,6434],{},"断言",[57,6436,6438,6480],{"className":6437},[60],[57,6439,6441],{"className":6440},[64],[66,6442,6443],{"xmlns":68},[70,6444,6445,6477],{},[73,6446,6447,6453,6455,6461,6463,6465,6467,6473,6475],{},[80,6448,6449,6451],{},[83,6450,5084],{},[278,6452,280],{},[76,6454,1008],{},[80,6456,6457,6459],{},[83,6458,5084],{},[278,6460,290],{},[76,6462,1008],{},[76,6464,4483],{},[76,6466,1008],{},[80,6468,6469,6471],{},[83,6470,5084],{},[83,6472,2206],{},[76,6474,1030],{},[83,6476,85],{},[90,6478,6479],{"encoding":92},"M_1 + M_2 + \\dots + M_n = R",[57,6481,6483,6538,6593,6611,6666],{"className":6482,"ariaHidden":98},[97],[57,6484,6486,6489,6529,6532,6535],{"className":6485},[102],[57,6487],{"className":6488,"style":219},[106],[57,6490,6492,6495],{"className":6491},[123],[57,6493,5084],{"className":6494,"style":5105},[123,127],[57,6496,6498],{"className":6497},[132],[57,6499,6501,6521],{"className":6500},[136,137],[57,6502,6504,6518],{"className":6503},[141],[57,6505,6507],{"className":6506,"style":339},[145],[57,6508,6509,6512],{"style":5120},[57,6510],{"className":6511,"style":154},[153],[57,6513,6515],{"className":6514},[158,159,160,161],[57,6516,280],{"className":6517},[123,161],[57,6519,169],{"className":6520},[168],[57,6522,6524],{"className":6523},[141],[57,6525,6527],{"className":6526,"style":176},[145],[57,6528],{},[57,6530],{"className":6531,"style":1172},[118],[57,6533,1008],{"className":6534},[1176],[57,6536],{"className":6537,"style":1172},[118],[57,6539,6541,6544,6584,6587,6590],{"className":6540},[102],[57,6542],{"className":6543,"style":219},[106],[57,6545,6547,6550],{"className":6546},[123],[57,6548,5084],{"className":6549,"style":5105},[123,127],[57,6551,6553],{"className":6552},[132],[57,6554,6556,6576],{"className":6555},[136,137],[57,6557,6559,6573],{"className":6558},[141],[57,6560,6562],{"className":6561,"style":339},[145],[57,6563,6564,6567],{"style":5120},[57,6565],{"className":6566,"style":154},[153],[57,6568,6570],{"className":6569},[158,159,160,161],[57,6571,290],{"className":6572},[123,161],[57,6574,169],{"className":6575},[168],[57,6577,6579],{"className":6578},[141],[57,6580,6582],{"className":6581,"style":176},[145],[57,6583],{},[57,6585],{"className":6586,"style":1172},[118],[57,6588,1008],{"className":6589},[1176],[57,6591],{"className":6592,"style":1172},[118],[57,6594,6596,6599,6602,6605,6608],{"className":6595},[102],[57,6597],{"className":6598,"style":3129},[106],[57,6600,4483],{"className":6601},[672],[57,6603],{"className":6604,"style":1172},[118],[57,6606,1008],{"className":6607},[1176],[57,6609],{"className":6610,"style":1172},[118],[57,6612,6614,6617,6657,6660,6663],{"className":6613},[102],[57,6615],{"className":6616,"style":219},[106],[57,6618,6620,6623],{"className":6619},[123],[57,6621,5084],{"className":6622,"style":5105},[123,127],[57,6624,6626],{"className":6625},[132],[57,6627,6629,6649],{"className":6628},[136,137],[57,6630,6632,6646],{"className":6631},[141],[57,6633,6635],{"className":6634,"style":2295},[145],[57,6636,6637,6640],{"style":5120},[57,6638],{"className":6639,"style":154},[153],[57,6641,6643],{"className":6642},[158,159,160,161],[57,6644,2206],{"className":6645},[123,127,161],[57,6647,169],{"className":6648},[168],[57,6650,6652],{"className":6651},[141],[57,6653,6655],{"className":6654,"style":176},[145],[57,6656],{},[57,6658],{"className":6659,"style":1281},[118],[57,6661,1030],{"className":6662},[1285],[57,6664],{"className":6665,"style":1281},[118],[57,6667,6669,6672],{"className":6668},[102],[57,6670],{"className":6671,"style":2507},[106],[57,6673,85],{"className":6674,"style":128},[123,127],"。\n课件用了一个复杂的归纳法来证明这一点。目的是为了说明这些“乘积”加起来能覆盖整个环，特别是能造出 ",[24,6677,280],{},[6679,6680],"br",{},[24,6682,6683],{},"归纳法拆解：",[184,6685,6686,6984,7221,7550,8464],{},[21,6687,6688,6689,2879,6759,6828,6829,459],{},"我们知道 ",[57,6690,6692,6710],{"className":6691},[60],[57,6693,6695],{"className":6694},[64],[66,6696,6697],{"xmlns":68},[70,6698,6699,6707],{},[73,6700,6701],{},[80,6702,6703,6705],{},[83,6704,2354],{},[83,6706,88],{},[90,6708,6709],{"encoding":92},"I_i",[57,6711,6713],{"className":6712,"ariaHidden":98},[97],[57,6714,6716,6719],{"className":6715},[102],[57,6717],{"className":6718,"style":219},[106],[57,6720,6722,6725],{"className":6721},[123],[57,6723,2354],{"className":6724,"style":2387},[123,127],[57,6726,6728],{"className":6727},[132],[57,6729,6731,6751],{"className":6730},[136,137],[57,6732,6734,6748],{"className":6733},[141],[57,6735,6737],{"className":6736,"style":146},[145],[57,6738,6739,6742],{"style":2402},[57,6740],{"className":6741,"style":154},[153],[57,6743,6745],{"className":6744},[158,159,160,161],[57,6746,88],{"className":6747},[123,127,161],[57,6749,169],{"className":6750},[168],[57,6752,6754],{"className":6753},[141],[57,6755,6757],{"className":6756,"style":176},[145],[57,6758],{},[57,6760,6762,6779],{"className":6761},[60],[57,6763,6765],{"className":6764},[64],[66,6766,6767],{"xmlns":68},[70,6768,6769,6777],{},[73,6770,6771],{},[80,6772,6773,6775],{},[83,6774,2354],{},[83,6776,5170],{},[90,6778,6379],{"encoding":92},[57,6780,6782],{"className":6781,"ariaHidden":98},[97],[57,6783,6785,6788],{"className":6784},[102],[57,6786],{"className":6787,"style":6317},[106],[57,6789,6791,6794],{"className":6790},[123],[57,6792,2354],{"className":6793,"style":2387},[123,127],[57,6795,6797],{"className":6796},[132],[57,6798,6800,6820],{"className":6799},[136,137],[57,6801,6803,6817],{"className":6802},[141],[57,6804,6806],{"className":6805,"style":146},[145],[57,6807,6808,6811],{"style":2402},[57,6809],{"className":6810,"style":154},[153],[57,6812,6814],{"className":6813},[158,159,160,161],[57,6815,5170],{"className":6816,"style":5283},[123,127,161],[57,6818,169],{"className":6819},[168],[57,6821,6823],{"className":6822},[141],[57,6824,6826],{"className":6825,"style":5383},[145],[57,6827],{}," 是互素的（",[57,6830,6832,6862],{"className":6831},[60],[57,6833,6835],{"className":6834},[64],[66,6836,6837],{"xmlns":68},[70,6838,6839,6859],{},[73,6840,6841,6847,6849,6855,6857],{},[80,6842,6843,6845],{},[83,6844,2354],{},[83,6846,88],{},[76,6848,1008],{},[80,6850,6851,6853],{},[83,6852,2354],{},[83,6854,5170],{},[76,6856,1030],{},[83,6858,85],{},[90,6860,6861],{"encoding":92},"I_i + I_j = R",[57,6863,6865,6920,6975],{"className":6864,"ariaHidden":98},[97],[57,6866,6868,6871,6911,6914,6917],{"className":6867},[102],[57,6869],{"className":6870,"style":219},[106],[57,6872,6874,6877],{"className":6873},[123],[57,6875,2354],{"className":6876,"style":2387},[123,127],[57,6878,6880],{"className":6879},[132],[57,6881,6883,6903],{"className":6882},[136,137],[57,6884,6886,6900],{"className":6885},[141],[57,6887,6889],{"className":6888,"style":146},[145],[57,6890,6891,6894],{"style":2402},[57,6892],{"className":6893,"style":154},[153],[57,6895,6897],{"className":6896},[158,159,160,161],[57,6898,88],{"className":6899},[123,127,161],[57,6901,169],{"className":6902},[168],[57,6904,6906],{"className":6905},[141],[57,6907,6909],{"className":6908,"style":176},[145],[57,6910],{},[57,6912],{"className":6913,"style":1172},[118],[57,6915,1008],{"className":6916},[1176],[57,6918],{"className":6919,"style":1172},[118],[57,6921,6923,6926,6966,6969,6972],{"className":6922},[102],[57,6924],{"className":6925,"style":6317},[106],[57,6927,6929,6932],{"className":6928},[123],[57,6930,2354],{"className":6931,"style":2387},[123,127],[57,6933,6935],{"className":6934},[132],[57,6936,6938,6958],{"className":6937},[136,137],[57,6939,6941,6955],{"className":6940},[141],[57,6942,6944],{"className":6943,"style":146},[145],[57,6945,6946,6949],{"style":2402},[57,6947],{"className":6948,"style":154},[153],[57,6950,6952],{"className":6951},[158,159,160,161],[57,6953,5170],{"className":6954,"style":5283},[123,127,161],[57,6956,169],{"className":6957},[168],[57,6959,6961],{"className":6960},[141],[57,6962,6964],{"className":6963,"style":5383},[145],[57,6965],{},[57,6967],{"className":6968,"style":1281},[118],[57,6970,1030],{"className":6971},[1285],[57,6973],{"className":6974,"style":1281},[118],[57,6976,6978,6981],{"className":6977},[102],[57,6979],{"className":6980,"style":2507},[106],[57,6982,85],{"className":6983,"style":128},[123,127],[21,6985,6986,6989,6990,7066,7067,7142,7143,3906],{},[24,6987,6988],{},"课件引用的命题 3.3.5"," 其实是一个性质：如果 ",[57,6991,6993,7017],{"className":6992},[60],[57,6994,6996],{"className":6995},[64],[66,6997,6998],{"xmlns":68},[70,6999,7000,7014],{},[73,7001,7002,7005,7007,7010,7012],{},[83,7003,7004],{},"A",[76,7006,1008],{},[83,7008,7009],{},"B",[76,7011,1030],{},[83,7013,85],{},[90,7015,7016],{"encoding":92},"A+B=R",[57,7018,7020,7038,7057],{"className":7019,"ariaHidden":98},[97],[57,7021,7023,7026,7029,7032,7035],{"className":7022},[102],[57,7024],{"className":7025,"style":2945},[106],[57,7027,7004],{"className":7028},[123,127],[57,7030],{"className":7031,"style":1172},[118],[57,7033,1008],{"className":7034},[1176],[57,7036],{"className":7037,"style":1172},[118],[57,7039,7041,7044,7048,7051,7054],{"className":7040},[102],[57,7042],{"className":7043,"style":2507},[106],[57,7045,7009],{"className":7046,"style":7047},[123,127],"margin-right:0.0502em;",[57,7049],{"className":7050,"style":1281},[118],[57,7052,1030],{"className":7053},[1285],[57,7055],{"className":7056,"style":1281},[118],[57,7058,7060,7063],{"className":7059},[102],[57,7061],{"className":7062,"style":2507},[106],[57,7064,85],{"className":7065,"style":128},[123,127]," 且 ",[57,7068,7070,7093],{"className":7069},[60],[57,7071,7073],{"className":7072},[64],[66,7074,7075],{"xmlns":68},[70,7076,7077,7090],{},[73,7078,7079,7081,7083,7086,7088],{},[83,7080,7004],{},[76,7082,1008],{},[83,7084,7085],{},"C",[76,7087,1030],{},[83,7089,85],{},[90,7091,7092],{"encoding":92},"A+C=R",[57,7094,7096,7114,7133],{"className":7095,"ariaHidden":98},[97],[57,7097,7099,7102,7105,7108,7111],{"className":7098},[102],[57,7100],{"className":7101,"style":2945},[106],[57,7103,7004],{"className":7104},[123,127],[57,7106],{"className":7107,"style":1172},[118],[57,7109,1008],{"className":7110},[1176],[57,7112],{"className":7113,"style":1172},[118],[57,7115,7117,7120,7124,7127,7130],{"className":7116},[102],[57,7118],{"className":7119,"style":2507},[106],[57,7121,7085],{"className":7122,"style":7123},[123,127],"margin-right:0.0715em;",[57,7125],{"className":7126,"style":1281},[118],[57,7128,1030],{"className":7129},[1285],[57,7131],{"className":7132,"style":1281},[118],[57,7134,7136,7139],{"className":7135},[102],[57,7137],{"className":7138,"style":2507},[106],[57,7140,85],{"className":7141,"style":128},[123,127],"，那么 ",[57,7144,7146,7170],{"className":7145},[60],[57,7147,7149],{"className":7148},[64],[66,7150,7151],{"xmlns":68},[70,7152,7153,7167],{},[73,7154,7155,7157,7159,7161,7163,7165],{},[83,7156,7004],{},[76,7158,1008],{},[83,7160,7009],{},[83,7162,7085],{},[76,7164,1030],{},[83,7166,85],{},[90,7168,7169],{"encoding":92},"A+BC=R",[57,7171,7173,7191,7212],{"className":7172,"ariaHidden":98},[97],[57,7174,7176,7179,7182,7185,7188],{"className":7175},[102],[57,7177],{"className":7178,"style":2945},[106],[57,7180,7004],{"className":7181},[123,127],[57,7183],{"className":7184,"style":1172},[118],[57,7186,1008],{"className":7187},[1176],[57,7189],{"className":7190,"style":1172},[118],[57,7192,7194,7197,7200,7203,7206,7209],{"className":7193},[102],[57,7195],{"className":7196,"style":2507},[106],[57,7198,7009],{"className":7199,"style":7047},[123,127],[57,7201,7085],{"className":7202,"style":7123},[123,127],[57,7204],{"className":7205,"style":1281},[118],[57,7207,1030],{"className":7208},[1285],[57,7210],{"className":7211,"style":1281},[118],[57,7213,7215,7218],{"className":7214},[102],[57,7216],{"className":7217,"style":2507},[106],[57,7219,85],{"className":7220,"style":128},[123,127],[21,7222,7223,7226,7227,7519,7520,7549],{},[24,7224,7225],{},"归纳假设","：假设 ",[57,7228,7230,7282],{"className":7229},[60],[57,7231,7233],{"className":7232},[64],[66,7234,7235],{"xmlns":68},[70,7236,7237,7279],{},[73,7238,7239,7245,7247,7249,7251,7257,7259,7271,7273],{},[80,7240,7241,7243],{},[83,7242,5084],{},[278,7244,280],{},[76,7246,1008],{},[76,7248,4483],{},[76,7250,1008],{},[80,7252,7253,7255],{},[83,7254,5084],{},[83,7256,2545],{},[76,7258,1030],{},[80,7260,7261,7263],{},[83,7262,2354],{},[73,7264,7265,7267,7269],{},[83,7266,2545],{},[76,7268,1008],{},[278,7270,280],{},[76,7272,4483],{},[80,7274,7275,7277],{},[83,7276,2354],{},[83,7278,2206],{},[90,7280,7281],{"encoding":92},"M_1 + \\dots + M_k = I_{k+1} \\cdots I_n",[57,7283,7285,7340,7358,7413],{"className":7284,"ariaHidden":98},[97],[57,7286,7288,7291,7331,7334,7337],{"className":7287},[102],[57,7289],{"className":7290,"style":219},[106],[57,7292,7294,7297],{"className":7293},[123],[57,7295,5084],{"className":7296,"style":5105},[123,127],[57,7298,7300],{"className":7299},[132],[57,7301,7303,7323],{"className":7302},[136,137],[57,7304,7306,7320],{"className":7305},[141],[57,7307,7309],{"className":7308,"style":339},[145],[57,7310,7311,7314],{"style":5120},[57,7312],{"className":7313,"style":154},[153],[57,7315,7317],{"className":7316},[158,159,160,161],[57,7318,280],{"className":7319},[123,161],[57,7321,169],{"className":7322},[168],[57,7324,7326],{"className":7325},[141],[57,7327,7329],{"className":7328,"style":176},[145],[57,7330],{},[57,7332],{"className":7333,"style":1172},[118],[57,7335,1008],{"className":7336},[1176],[57,7338],{"className":7339,"style":1172},[118],[57,7341,7343,7346,7349,7352,7355],{"className":7342},[102],[57,7344],{"className":7345,"style":3129},[106],[57,7347,4483],{"className":7348},[672],[57,7350],{"className":7351,"style":1172},[118],[57,7353,1008],{"className":7354},[1176],[57,7356],{"className":7357,"style":1172},[118],[57,7359,7361,7364,7404,7407,7410],{"className":7360},[102],[57,7362],{"className":7363,"style":219},[106],[57,7365,7367,7370],{"className":7366},[123],[57,7368,5084],{"className":7369,"style":5105},[123,127],[57,7371,7373],{"className":7372},[132],[57,7374,7376,7396],{"className":7375},[136,137],[57,7377,7379,7393],{"className":7378},[141],[57,7380,7382],{"className":7381,"style":2695},[145],[57,7383,7384,7387],{"style":5120},[57,7385],{"className":7386,"style":154},[153],[57,7388,7390],{"className":7389},[158,159,160,161],[57,7391,2545],{"className":7392,"style":2643},[123,127,161],[57,7394,169],{"className":7395},[168],[57,7397,7399],{"className":7398},[141],[57,7400,7402],{"className":7401,"style":176},[145],[57,7403],{},[57,7405],{"className":7406,"style":1281},[118],[57,7408,1030],{"className":7409},[1285],[57,7411],{"className":7412,"style":1281},[118],[57,7414,7416,7420,7470,7473,7476,7479],{"className":7415},[102],[57,7417],{"className":7418,"style":7419},[106],"height:0.8917em;vertical-align:-0.2083em;",[57,7421,7423,7426],{"className":7422},[123],[57,7424,2354],{"className":7425,"style":2387},[123,127],[57,7427,7429],{"className":7428},[132],[57,7430,7432,7461],{"className":7431},[136,137],[57,7433,7435,7458],{"className":7434},[141],[57,7436,7438],{"className":7437,"style":2695},[145],[57,7439,7440,7443],{"style":2402},[57,7441],{"className":7442,"style":154},[153],[57,7444,7446],{"className":7445},[158,159,160,161],[57,7447,7449,7452,7455],{"className":7448},[123,161],[57,7450,2545],{"className":7451,"style":2643},[123,127,161],[57,7453,1008],{"className":7454},[1176,161],[57,7456,280],{"className":7457},[123,161],[57,7459,169],{"className":7460},[168],[57,7462,7464],{"className":7463},[141],[57,7465,7468],{"className":7466,"style":7467},[145],"height:0.2083em;",[57,7469],{},[57,7471],{"className":7472,"style":119},[118],[57,7474,4483],{"className":7475},[672],[57,7477],{"className":7478,"style":119},[118],[57,7480,7482,7485],{"className":7481},[123],[57,7483,2354],{"className":7484,"style":2387},[123,127],[57,7486,7488],{"className":7487},[132],[57,7489,7491,7511],{"className":7490},[136,137],[57,7492,7494,7508],{"className":7493},[141],[57,7495,7497],{"className":7496,"style":2295},[145],[57,7498,7499,7502],{"style":2402},[57,7500],{"className":7501,"style":154},[153],[57,7503,7505],{"className":7504},[158,159,160,161],[57,7506,2206],{"className":7507},[123,127,161],[57,7509,169],{"className":7510},[168],[57,7512,7514],{"className":7513},[141],[57,7515,7517],{"className":7516,"style":176},[145],[57,7518],{}," （课件这里写得非常简略，其实意思是前 ",[57,7521,7523,7536],{"className":7522},[60],[57,7524,7526],{"className":7525},[64],[66,7527,7528],{"xmlns":68},[70,7529,7530,7534],{},[73,7531,7532],{},[83,7533,2545],{},[90,7535,2545],{"encoding":92},[57,7537,7539],{"className":7538,"ariaHidden":98},[97],[57,7540,7542,7546],{"className":7541},[102],[57,7543],{"className":7544,"style":7545},[106],"height:0.6944em;",[57,7547,2545],{"className":7548,"style":2643},[123,127]," 项的和，包含了后面剩余项的乘积，这里有笔误或者是符号借用，我们直接看它的核心推导）。",[21,7551,7552,5141,7555],{},[24,7553,7554],{},"核心推导",[57,7556,7559],{"className":7557},[7558],"katex-display",[57,7560,7562,7734],{"className":7561},[60],[57,7563,7565],{"className":7564},[64],[66,7566,7568],{"xmlns":68,"display":7567},"block",[70,7569,7570,7731],{},[7571,7572,7576,7660],"mtable",{"rowspacing":7573,"columnalign":7574,"columnspacing":7575},"0.25em","right left","0em",[7577,7578,7579,7605],"mtr",{},[7580,7581,7582],"mtd",{},[7583,7584,7585],"mstyle",{"scriptlevel":3929,"displaystyle":98},[73,7586,7587,7593,7595,7597,7599],{},[80,7588,7589,7591],{},[83,7590,5084],{},[278,7592,280],{},[76,7594,1008],{},[76,7596,4483],{},[76,7598,1008],{},[80,7600,7601,7603],{},[83,7602,5084],{},[83,7604,2545],{},[7580,7606,7607],{},[7583,7608,7609],{"scriptlevel":3929,"displaystyle":98},[73,7610,7611,7613,7615,7621,7623,7629,7631,7644,7650,7653,7655,7658],{},[73,7612],{},[76,7614,1030],{},[80,7616,7617,7619],{},[83,7618,2354],{},[83,7620,2545],{},[76,7622,4483],{},[80,7624,7625,7627],{},[83,7626,2354],{},[83,7628,2206],{},[76,7630,1008],{},[7632,7633,7634,7636],"munder",{},[76,7635,78],{},[73,7637,7638,7640,7642],{},[83,7639,5170],{},[76,7641,5173],{"mathvariant":302},[83,7643,2545],{},[80,7645,7646,7648],{},[83,7647,2354],{},[83,7649,5170],{},[118,7651],{"width":7652},"1em",[76,7654,478],{"stretchy":477},[508,7656,7657],{},"这一步是在拆分",[76,7659,513],{"stretchy":477},[7577,7661,7662,7668],{},[7580,7663,7664],{},[7583,7665,7666],{"scriptlevel":3929,"displaystyle":98},[73,7667],{},[7580,7669,7670],{},[7583,7671,7672],{"scriptlevel":3929,"displaystyle":98},[73,7673,7674,7676,7678,7680,7686,7688,7694,7696,7709,7711,7723,7725],{},[73,7675],{},[76,7677,1030],{},[76,7679,478],{"stretchy":477},[80,7681,7682,7684],{},[83,7683,2354],{},[83,7685,2545],{},[76,7687,1008],{},[80,7689,7690,7692],{},[83,7691,2354],{},[278,7693,280],{},[76,7695,4483],{},[80,7697,7698,7700],{},[83,7699,2354],{},[73,7701,7702,7704,7707],{},[83,7703,2545],{},[76,7705,7706],{},"−",[278,7708,280],{},[76,7710,513],{"stretchy":477},[80,7712,7713,7715],{},[83,7714,2354],{},[73,7716,7717,7719,7721],{},[83,7718,2545],{},[76,7720,1008],{},[278,7722,280],{},[76,7724,4483],{},[80,7726,7727,7729],{},[83,7728,2354],{},[83,7730,2206],{},[90,7732,7733],{"encoding":92},"\\begin{aligned}\nM_1 + \\dots + M_k &= I_{k} \\cdots I_n + \\prod_{j \\neq k} I_j \\quad (\\text{这一步是在拆分}) \\\\\n&= (I_k + I_1 \\cdots I_{k-1}) I_{k+1} \\cdots I_n\n\\end{aligned}",[57,7735,7737],{"className":7736,"ariaHidden":98},[97],[57,7738,7740,7744],{"className":7739},[102],[57,7741],{"className":7742,"style":7743},[106],"height:3.9882em;vertical-align:-1.7441em;",[57,7745,7747],{"className":7746},[123],[57,7748,7750,7896],{"className":7749},[7571],[57,7751,7754],{"className":7752},[7753],"col-align-r",[57,7755,7757,7887],{"className":7756},[136,137],[57,7758,7760,7884],{"className":7759},[141],[57,7761,7764,7875],{"className":7762,"style":7763},[145],"height:2.2441em;",[57,7765,7767,7771],{"style":7766},"top:-4.2441em;",[57,7768],{"className":7769,"style":7770},[153],"height:3.05em;",[57,7772,7774,7814,7817,7820,7823,7826,7829,7832,7835],{"className":7773},[123],[57,7775,7777,7780],{"className":7776},[123],[57,7778,5084],{"className":7779,"style":5105},[123,127],[57,7781,7783],{"className":7782},[132],[57,7784,7786,7806],{"className":7785},[136,137],[57,7787,7789,7803],{"className":7788},[141],[57,7790,7792],{"className":7791,"style":339},[145],[57,7793,7794,7797],{"style":5120},[57,7795],{"className":7796,"style":154},[153],[57,7798,7800],{"className":7799},[158,159,160,161],[57,7801,280],{"className":7802},[123,161],[57,7804,169],{"className":7805},[168],[57,7807,7809],{"className":7808},[141],[57,7810,7812],{"className":7811,"style":176},[145],[57,7813],{},[57,7815],{"className":7816,"style":1172},[118],[57,7818,1008],{"className":7819},[1176],[57,7821],{"className":7822,"style":1172},[118],[57,7824,4483],{"className":7825},[672],[57,7827],{"className":7828,"style":1172},[118],[57,7830,1008],{"className":7831},[1176],[57,7833],{"className":7834,"style":1172},[118],[57,7836,7838,7841],{"className":7837},[123],[57,7839,5084],{"className":7840,"style":5105},[123,127],[57,7842,7844],{"className":7843},[132],[57,7845,7847,7867],{"className":7846},[136,137],[57,7848,7850,7864],{"className":7849},[141],[57,7851,7853],{"className":7852,"style":2695},[145],[57,7854,7855,7858],{"style":5120},[57,7856],{"className":7857,"style":154},[153],[57,7859,7861],{"className":7860},[158,159,160,161],[57,7862,2545],{"className":7863,"style":2643},[123,127,161],[57,7865,169],{"className":7866},[168],[57,7868,7870],{"className":7869},[141],[57,7871,7873],{"className":7872,"style":176},[145],[57,7874],{},[57,7876,7878,7881],{"style":7877},"top:-1.6659em;",[57,7879],{"className":7880,"style":7770},[153],[57,7882],{"className":7883},[123],[57,7885,169],{"className":7886},[168],[57,7888,7890],{"className":7889},[141],[57,7891,7894],{"className":7892,"style":7893},[145],"height:1.7441em;",[57,7895],{},[57,7897,7900],{"className":7898},[7899],"col-align-l",[57,7901,7903,8456],{"className":7902},[136,137],[57,7904,7906,8453],{"className":7905},[141],[57,7907,7909,8182],{"className":7908,"style":7763},[145],[57,7910,7911,7914],{"style":7766},[57,7912],{"className":7913,"style":7770},[153],[57,7915,7917,7920,7923,7926,7929,7972,7975,7978,7981,8021,8024,8027,8030,8122,8125,8165,8169,8172,8179],{"className":7916},[123],[57,7918],{"className":7919},[123],[57,7921],{"className":7922,"style":1281},[118],[57,7924,1030],{"className":7925},[1285],[57,7927],{"className":7928,"style":1281},[118],[57,7930,7932,7935],{"className":7931},[123],[57,7933,2354],{"className":7934,"style":2387},[123,127],[57,7936,7938],{"className":7937},[132],[57,7939,7941,7964],{"className":7940},[136,137],[57,7942,7944,7961],{"className":7943},[141],[57,7945,7947],{"className":7946,"style":2695},[145],[57,7948,7949,7952],{"style":2402},[57,7950],{"className":7951,"style":154},[153],[57,7953,7955],{"className":7954},[158,159,160,161],[57,7956,7958],{"className":7957},[123,161],[57,7959,2545],{"className":7960,"style":2643},[123,127,161],[57,7962,169],{"className":7963},[168],[57,7965,7967],{"className":7966},[141],[57,7968,7970],{"className":7969,"style":176},[145],[57,7971],{},[57,7973],{"className":7974,"style":119},[118],[57,7976,4483],{"className":7977},[672],[57,7979],{"className":7980,"style":119},[118],[57,7982,7984,7987],{"className":7983},[123],[57,7985,2354],{"className":7986,"style":2387},[123,127],[57,7988,7990],{"className":7989},[132],[57,7991,7993,8013],{"className":7992},[136,137],[57,7994,7996,8010],{"className":7995},[141],[57,7997,7999],{"className":7998,"style":2295},[145],[57,8000,8001,8004],{"style":2402},[57,8002],{"className":8003,"style":154},[153],[57,8005,8007],{"className":8006},[158,159,160,161],[57,8008,2206],{"className":8009},[123,127,161],[57,8011,169],{"className":8012},[168],[57,8014,8016],{"className":8015},[141],[57,8017,8019],{"className":8018,"style":176},[145],[57,8020],{},[57,8022],{"className":8023,"style":1172},[118],[57,8025,1008],{"className":8026},[1176],[57,8028],{"className":8029,"style":1172},[118],[57,8031,8034],{"className":8032},[111,8033],"op-limits",[57,8035,8037,8113],{"className":8036},[136,137],[57,8038,8040,8110],{"className":8039},[141],[57,8041,8044,8098],{"className":8042,"style":8043},[145],"height:1.05em;",[57,8045,8047,8050],{"style":8046},"top:-1.8479em;margin-left:0em;",[57,8048],{"className":8049,"style":7770},[153],[57,8051,8053],{"className":8052},[158,159,160,161],[57,8054,8056,8059,8095],{"className":8055},[123,161],[57,8057,5170],{"className":8058,"style":5283},[123,127,161],[57,8060,8062,8089,8092],{"className":8061},[1285,161],[57,8063,8065],{"className":8064},[1285,161],[57,8066,8068],{"className":8067},[123,5293,161],[57,8069,8071],{"className":8070},[5297,161],[57,8072,8074,8077,8086],{"className":8073},[5301,161],[57,8075],{"className":8076,"style":1353},[106],[57,8078,8080],{"className":8079},[5308],[57,8081,8083],{"className":8082},[123,161],[57,8084,5315],{"className":8085},[1285,161],[57,8087],{"className":8088},[5319],[57,8090],{"className":8091},[118,5323,161],[57,8093,1030],{"className":8094},[1285,161],[57,8096,2545],{"className":8097,"style":2643},[123,127,161],[57,8099,8101,8104],{"style":8100},"top:-3.05em;",[57,8102],{"className":8103,"style":7770},[153],[57,8105,8106],{},[57,8107,78],{"className":8108},[111,112,8109],"large-op",[57,8111,169],{"className":8112},[168],[57,8114,8116],{"className":8115},[141],[57,8117,8120],{"className":8118,"style":8119},[145],"height:1.4382em;",[57,8121],{},[57,8123],{"className":8124,"style":119},[118],[57,8126,8128,8131],{"className":8127},[123],[57,8129,2354],{"className":8130,"style":2387},[123,127],[57,8132,8134],{"className":8133},[132],[57,8135,8137,8157],{"className":8136},[136,137],[57,8138,8140,8154],{"className":8139},[141],[57,8141,8143],{"className":8142,"style":146},[145],[57,8144,8145,8148],{"style":2402},[57,8146],{"className":8147,"style":154},[153],[57,8149,8151],{"className":8150},[158,159,160,161],[57,8152,5170],{"className":8153,"style":5283},[123,127,161],[57,8155,169],{"className":8156},[168],[57,8158,8160],{"className":8159},[141],[57,8161,8163],{"className":8162,"style":5383},[145],[57,8164],{},[57,8166],{"className":8167,"style":8168},[118],"margin-right:1em;",[57,8170,478],{"className":8171},[529],[57,8173,8175],{"className":8174},[123,4851],[57,8176,7657],{"className":8177},[123,8178],"cjk_fallback",[57,8180,513],{"className":8181},[679],[57,8183,8184,8187],{"style":7877},[57,8185],{"className":8186,"style":7770},[153],[57,8188,8190,8193,8196,8199,8202,8205,8245,8248,8251,8254,8294,8297,8300,8303,8352,8355,8404,8407,8410,8413],{"className":8189},[123],[57,8191],{"className":8192},[123],[57,8194],{"className":8195,"style":1281},[118],[57,8197,1030],{"className":8198},[1285],[57,8200],{"className":8201,"style":1281},[118],[57,8203,478],{"className":8204},[529],[57,8206,8208,8211],{"className":8207},[123],[57,8209,2354],{"className":8210,"style":2387},[123,127],[57,8212,8214],{"className":8213},[132],[57,8215,8217,8237],{"className":8216},[136,137],[57,8218,8220,8234],{"className":8219},[141],[57,8221,8223],{"className":8222,"style":2695},[145],[57,8224,8225,8228],{"style":2402},[57,8226],{"className":8227,"style":154},[153],[57,8229,8231],{"className":8230},[158,159,160,161],[57,8232,2545],{"className":8233,"style":2643},[123,127,161],[57,8235,169],{"className":8236},[168],[57,8238,8240],{"className":8239},[141],[57,8241,8243],{"className":8242,"style":176},[145],[57,8244],{},[57,8246],{"className":8247,"style":1172},[118],[57,8249,1008],{"className":8250},[1176],[57,8252],{"className":8253,"style":1172},[118],[57,8255,8257,8260],{"className":8256},[123],[57,8258,2354],{"className":8259,"style":2387},[123,127],[57,8261,8263],{"className":8262},[132],[57,8264,8266,8286],{"className":8265},[136,137],[57,8267,8269,8283],{"className":8268},[141],[57,8270,8272],{"className":8271,"style":339},[145],[57,8273,8274,8277],{"style":2402},[57,8275],{"className":8276,"style":154},[153],[57,8278,8280],{"className":8279},[158,159,160,161],[57,8281,280],{"className":8282},[123,161],[57,8284,169],{"className":8285},[168],[57,8287,8289],{"className":8288},[141],[57,8290,8292],{"className":8291,"style":176},[145],[57,8293],{},[57,8295],{"className":8296,"style":119},[118],[57,8298,4483],{"className":8299},[672],[57,8301],{"className":8302,"style":119},[118],[57,8304,8306,8309],{"className":8305},[123],[57,8307,2354],{"className":8308,"style":2387},[123,127],[57,8310,8312],{"className":8311},[132],[57,8313,8315,8344],{"className":8314},[136,137],[57,8316,8318,8341],{"className":8317},[141],[57,8319,8321],{"className":8320,"style":2695},[145],[57,8322,8323,8326],{"style":2402},[57,8324],{"className":8325,"style":154},[153],[57,8327,8329],{"className":8328},[158,159,160,161],[57,8330,8332,8335,8338],{"className":8331},[123,161],[57,8333,2545],{"className":8334,"style":2643},[123,127,161],[57,8336,7706],{"className":8337},[1176,161],[57,8339,280],{"className":8340},[123,161],[57,8342,169],{"className":8343},[168],[57,8345,8347],{"className":8346},[141],[57,8348,8350],{"className":8349,"style":7467},[145],[57,8351],{},[57,8353,513],{"className":8354},[679],[57,8356,8358,8361],{"className":8357},[123],[57,8359,2354],{"className":8360,"style":2387},[123,127],[57,8362,8364],{"className":8363},[132],[57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assName":8699,"style":1281},[118],[57,8701,8703,8706],{"className":8702},[102],[57,8704],{"className":8705,"style":2507},[106],[57,8707,85],{"className":8708,"style":128},[123,127],[21,8710,8711,8714,8715,8816,8817,8819,8820,8954,8955,9196,9198,9201,11000,11002,55,11076],{},[24,8712,8713],{},"构造\"魔法开关\"：","\n既然 ",[57,8716,8718,8743],{"className":8717},[60],[57,8719,8721],{"className":8720},[64],[66,8722,8723],{"xmlns":68},[70,8724,8725,8740],{},[73,8726,8727,8730,8736,8738],{},[76,8728,8729],{},"∑",[80,8731,8732,8734],{},[83,8733,5084],{},[83,8735,88],{},[76,8737,1030],{},[83,8739,85],{},[90,8741,8742],{"encoding":92},"\\sum M_i = R",[57,8744,8746,8807],{"className":8745,"ariaHidden":98},[97],[57,8747,8749,8752,8755,8758,8798,8801,8804],{"className":8748},[102],[57,8750],{"className":8751,"style":107},[106],[57,8753,8729],{"className":8754,"style":114},[111,112,113],[57,8756],{"className":8757,"style":119},[118],[57,8759,8761,8764],{"className":8760},[123],[57,8762,5084],{"className":8763,"style":5105},[123,127],[57,8765,8767],{"className":8766},[132],[57,8768,8770,8790],{"className":8769},[136,137],[57,8771,8773,8787],{"className":8772},[141],[57,8774,8776],{"className":8775,"style":146},[145],[57,8777,8778,8781],{"style":5120},[57,8779],{"className":8780,"style":154},[153],[57,8782,8784],{"className":8783},[158,159,160,161],[57,8785,88],{"className":8786},[123,127,161],[57,8788,169],{"className":8789},[168],[57,8791,8793],{"className":8792},[141],[57,8794,8796],{"className":8795,"style":176},[145],[57,8797],{},[57,8799],{"className":8800,"style":1281},[118],[57,8802,1030],{"className":8803},[1285],[57,8805],{"className":8806,"style":1281},[118],[57,8808,8810,8813],{"className":8809},[102],[57,8811],{"className":8812,"style":2507},[106],[57,8814,85],{"className":8815,"style":128},[123,127],"，那么单位元 ",[24,8818,280],{}," 也可以写成它们的和。\n存在 ",[57,8821,8823,8849],{"className":8822},[60],[57,8824,8826],{"className":8825},[64],[66,8827,8828],{"xmlns":68},[70,8829,8830,8846],{},[73,8831,8832,8838,8840],{},[80,8833,8834,8836],{},[83,8835,3941],{},[83,8837,88],{},[76,8839,3003],{},[80,8841,8842,8844],{},[83,8843,5084],{},[83,8845,88],{},[90,8847,8848],{"encoding":92},"m_i \\in M_i",[57,8850,8852,8908],{"className":8851,"ariaHidden":98},[97],[57,8853,8855,8859,8899,8902,8905],{"className":8854},[102],[57,8856],{"className":8857,"style":8858},[106],"height:0.6891em;vertical-align:-0.15em;",[57,8860,8862,8865],{"className":8861},[123],[57,8863,3941],{"className":8864},[123,127],[57,8866,8868],{"className":8867},[132],[57,8869,8871,8891],{"className":8870},[136,137],[57,8872,8874,8888],{"className":8873},[141],[57,8875,8877],{"className":8876,"style":146},[145],[57,8878,8879,8882],{"style":550},[57,8880],{"className":8881,"style":154},[153],[57,8883,8885],{"className":8884},[158,159,160,161],[57,8886,88],{"className":8887},[123,127,161],[57,8889,169],{"className":8890},[168],[57,8892,8894],{"className":8893},[141],[57,8895,8897],{"className":8896,"style":176},[145],[57,8898],{},[57,8900],{"className":8901,"style":1281},[118],[57,8903,3003],{"className":8904},[1285],[57,8906],{"className":8907,"style":1281},[118],[57,8909,8911,8914],{"className":8910},[102],[57,8912],{"className":8913,"style":219},[106],[57,8915,8917,8920],{"className":8916},[123],[57,8918,5084],{"className":8919,"style":5105},[123,127],[57,8921,8923],{"className":8922},[132],[57,8924,8926,8946],{"className":8925},[136,137],[57,8927,8929,8943],{"className":8928},[141],[57,8930,8932],{"className":8931,"style":146},[145],[57,8933,8934,8937],{"style":5120},[57,8935],{"className":8936,"style":154},[153],[57,8938,8940],{"className":8939},[158,159,160,161],[57,8941,88],{"className":8942},[123,127,161],[57,8944,169],{"className":8945},[168],[57,8947,8949],{"className":8948},[141],[57,8950,8952],{"className":8951,"style":176},[145],[57,8953],{},"，使得：\n",[57,8956,8958,9000],{"className":8957},[60],[57,8959,8961],{"className":8960},[64],[66,8962,8963],{"xmlns":68},[70,8964,8965,8997],{},[73,8966,8967,8969,8971,8977,8979,8985,8987,8989,8991],{},[278,8968,280],{},[76,8970,1030],{},[80,8972,8973,8975],{},[83,8974,3941],{},[278,8976,280],{},[76,8978,1008],{},[80,8980,8981,8983],{},[83,8982,3941],{},[278,8984,290],{},[76,8986,1008],{},[76,8988,4483],{},[76,8990,1008],{},[80,8992,8993,8995],{},[83,8994,3941],{},[83,8996,2206],{},[90,8998,8999],{"encoding":92},"1 = m_1 + m_2 + \\dots + m_n",[57,9001,9003,9021,9077,9132,9150],{"className":9002,"ariaHidden":98},[97],[57,9004,9006,9009,9012,9015,9018],{"className":9005},[102],[57,9007],{"className":9008,"style":3167},[106],[57,9010,280],{"className":9011},[123],[57,9013],{"className":9014,"style":1281},[118],[57,9016,1030],{"className":9017},[1285],[57,9019],{"className":9020,"style":1281},[118],[57,9022,9024,9028,9068,9071,9074],{"className":9023},[102],[57,9025],{"className":9026,"style":9027},[106],"height:0.7333em;vertical-align:-0.15em;",[57,9029,9031,9034],{"className":9030},[123],[57,9032,3941],{"className":9033},[123,127],[57,9035,9037],{"className":9036},[132],[57,9038,9040,9060],{"className":9039},[136,137],[57,9041,9043,9057],{"className":9042},[141],[57,9044,9046],{"className":9045,"style":339},[145],[57,9047,9048,9051],{"style":550},[57,9049],{"className":9050,"style":154},[153],[57,9052,9054],{"className":9053},[158,159,160,161],[57,9055,280],{"className":9056},[123,161],[57,9058,169],{"className":9059},[168],[57,9061,9063],{"className":9062},[141],[57,9064,9066],{"className":9065,"style":176},[145],[57,9067],{},[57,9069],{"className":9070,"style":1172},[118],[57,9072,1008],{"className":9073},[1176],[57,9075],{"className":9076,"style":1172},[118],[57,9078,9080,9083,9123,9126,9129],{"className":9079},[102],[57,9081],{"className":9082,"style":9027},[106],[57,9084,9086,9089],{"className":9085},[123],[57,9087,3941],{"className":9088},[123,127],[57,9090,9092],{"className":9091},[132],[57,9093,9095,9115],{"className":9094},[136,137],[57,9096,9098,9112],{"className":9097},[141],[57,9099,9101],{"className":9100,"style":339},[145],[57,9102,9103,9106],{"style":550},[57,9104],{"className":9105,"style":154},[153],[57,9107,9109],{"className":9108},[158,159,160,161],[57,9110,290],{"className":9111},[123,161],[57,9113,169],{"className":9114},[168],[57,9116,9118],{"className":9117},[141],[57,9119,9121],{"className":9120,"style":176},[145],[57,9122],{},[57,9124],{"className":9125,"style":1172},[118],[57,9127,1008],{"className":9128},[1176],[57,9130],{"className":9131,"style":1172},[118],[57,9133,9135,9138,9141,9144,9147],{"className":9134},[102],[57,9136],{"className":9137,"style":3129},[106],[57,9139,4483],{"className":9140},[672],[57,9142],{"className":9143,"style":1172},[118],[57,9145,1008],{"className":9146},[1176],[57,9148],{"className":9149,"style":1172},[118],[57,9151,9153,9156],{"className":9152},[102],[57,9154],{"className":9155,"style":711},[106],[57,9157,9159,9162],{"className":9158},[123],[57,9160,3941],{"className":9161},[123,127],[57,9163,9165],{"className":9164},[132],[57,9166,9168,9188],{"className":9167},[136,137],[57,9169,9171,9185],{"className":9170},[141],[57,9172,9174],{"className":9173,"style":2295},[145],[57,9175,9176,9179],{"style":550},[57,9177],{"className":9178,"style":154},[153],[57,9180,9182],{"className":9181},[158,159,160,161],[57,9183,2206],{"className":9184},[123,127,161],[57,9186,169],{"className":9187},[168],[57,9189,9191],{"className":9190},[141],[57,9192,9194],{"className":9193,"style":176},[145],[57,9195],{},[6679,9197],{},[24,9199,9200],{},"请盯着这个式子看，这是证明的灵魂：",[184,9202,9203,9750],{},[21,9204,9205,9206,9276,9277,9346,9347,5462,9416,9547,9548,9617,9618,9749],{},"考察 ",[57,9207,9209,9227],{"className":9208},[60],[57,9210,9212],{"className":9211},[64],[66,9213,9214],{"xmlns":68},[70,9215,9216,9224],{},[73,9217,9218],{},[80,9219,9220,9222],{},[83,9221,3941],{},[278,9223,280],{},[90,9225,9226],{"encoding":92},"m_1",[57,9228,9230],{"className":9229,"ariaHidden":98},[97],[57,9231,9233,9236],{"className":9232},[102],[57,9234],{"className":9235,"style":711},[106],[57,9237,9239,9242],{"className":9238},[123],[57,9240,3941],{"className":9241},[123,127],[57,9243,9245],{"className":9244},[132],[57,9246,9248,9268],{"className":9247},[136,137],[57,9249,9251,9265],{"className":9250},[141],[57,9252,9254],{"className":9253,"style":339},[145],[57,9255,9256,9259],{"style":550},[57,9257],{"className":9258,"style":154},[153],[57,9260,9262],{"className":9261},[158,159,160,161],[57,9263,280],{"className":9264},[123,161],[57,9266,169],{"className":9267},[168],[57,9269,9271],{"className":9270},[141],[57,9272,9274],{"className":9273,"style":176},[145],[57,9275],{},"：它属于 ",[57,9278,9280,9297],{"className":9279},[60],[57,9281,9283],{"className":9282},[64],[66,9284,9285],{"xmlns":68},[70,9286,9287,9295],{},[73,9288,9289],{},[80,9290,9291,9293],{},[83,9292,5084],{},[278,9294,280],{},[90,9296,5412],{"encoding":92},[57,9298,9300],{"className":9299,"ariaHidden":98},[97],[57,9301,9303,9306],{"className":9302},[102],[57,9304],{"className":9305,"style":219},[106],[57,9307,9309,9312],{"className":9308},[123],[57,9310,5084],{"className":9311,"style":5105},[123,127],[57,9313,9315],{"className":9314},[132],[57,9316,9318,9338],{"className":9317},[136,137],[57,9319,9321,9335],{"className":9320},[141],[57,9322,9324],{"className":9323,"style":339},[145],[57,9325,9326,9329],{"style":5120},[57,9327],{"className":9328,"style":154},[153],[57,9330,9332],{"className":9331},[158,159,160,161],[57,9333,280],{"className":9334},[123,161],[57,9336,169],{"className":9337},[168],[57,9339,9341],{"className":9340},[141],[57,9342,9344],{"className":9343,"style":176},[145],[57,9345],{},"。由定义，",[57,9348,9350,9367],{"className":9349},[60],[57,9351,9353],{"className":9352},[64],[66,9354,9355],{"xmlns":68},[70,9356,9357,9365],{},[73,9358,9359],{},[80,9360,9361,9363],{},[83,9362,5084],{},[278,9364,280],{},[90,9366,5412],{"encoding":92},[57,9368,9370],{"className":9369,"ariaHidden":98},[97],[57,9371,9373,9376],{"className":9372},[102],[57,9374],{"className":9375,"style":219},[106],[57,9377,9379,9382],{"className":9378},[123],[57,9380,5084],{"className":9381,"style":5105},[123,127],[57,9383,9385],{"className":9384},[132],[57,9386,9388,9408],{"className":9387},[136,137],[57,9389,9391,9405],{"className":9390},[141],[57,9392,9394],{"className":9393,"style":339},[145],[57,9395,9396,9399],{"style":5120},[57,9397],{"className":9398,"style":154},[153],[57,9400,9402],{"className":9401},[158,159,160,161],[57,9403,280],{"className":9404},[123,161],[57,9406,169],{"className":9407},[168],[57,9409,9411],{"className":9410},[141],[57,9412,9414],{"className":9413,"style":176},[145],[57,9415],{},[57,9417,9419,9446],{"className":9418},[60],[57,9420,9422],{"className":9421},[64],[66,9423,9424],{"xmlns":68},[70,9425,9426,9444],{},[73,9427,9428,9434,9436,9442],{},[80,9429,9430,9432],{},[83,9431,2354],{},[278,9433,290],{},[76,9435,283],{"separator":98},[80,9437,9438,9440],{},[83,9439,2354],{},[278,9441,299],{},[76,9443,506],{},[90,9445,5830],{"encoding":92},[57,9447,9449],{"className":9448,"ariaHidden":98},[97],[57,9450,9452,9455,9495,9498,9501,9541,9544],{"className":9451},[102],[57,9453],{"className":9454,"style":320},[106],[57,9456,9458,9461],{"className":9457},[123],[57,9459,2354],{"className":9460,"style":2387},[123,127],[57,9462,9464],{"className":9463},[132],[57,9465,9467,9487],{"className":9466},[136,137],[57,9468,9470,9484],{"className":9469},[141],[57,9471,9473],{"className":9472,"style":339},[145],[57,9474,9475,9478],{"style":2402},[57,9476],{"className":9477,"style":154},[153],[57,9479,9481],{"className":9480},[158,159,160,161],[57,9482,290],{"className":9483},[123,161],[57,9485,169],{"className":9486},[168],[57,9488,9490],{"className":9489},[141],[57,9491,9493],{"className":9492,"style":176},[145],[57,9494],{},[57,9496,283],{"className":9497},[365],[57,9499],{"className":9500,"style":119},[118],[57,9502,9504,9507],{"className":9503},[123],[57,9505,2354],{"className":9506,"style":2387},[123,127],[57,9508,9510],{"className":9509},[132],[57,9511,9513,9533],{"className":9512},[136,137],[57,9514,9516,9530],{"className":9515},[141],[57,9517,9519],{"className":9518,"style":339},[145],[57,9520,9521,9524],{"style":2402},[57,9522],{"className":9523,"style":154},[153],[57,9525,9527],{"className":9526},[158,159,160,161],[57,9528,299],{"className":9529},[123,161],[57,9531,169],{"className":9532},[168],[57,9534,9536],{"className":9535},[141],[57,9537,9539],{"className":9538,"style":176},[145],[57,9540],{},[57,9542],{"className":9543,"style":119},[118],[57,9545,506],{"className":9546},[672]," 的乘积。所以 ",[57,9549,9551,9568],{"className":9550},[60],[57,9552,9554],{"className":9553},[64],[66,9555,9556],{"xmlns":68},[70,9557,9558,9566],{},[73,9559,9560],{},[80,9561,9562,9564],{},[83,9563,3941],{},[278,9565,280],{},[90,9567,9226],{"encoding":92},[57,9569,9571],{"className":9570,"ariaHidden":98},[97],[57,9572,9574,9577],{"className":9573},[102],[57,9575],{"className":9576,"style":711},[106],[57,9578,9580,9583],{"className":9579},[123],[57,9581,3941],{"className":9582},[123,127],[57,9584,9586],{"className":9585},[132],[57,9587,9589,9609],{"className":9588},[136,137],[57,9590,9592,9606],{"className":9591},[141],[57,9593,9595],{"className":9594,"style":339},[145],[57,9596,9597,9600],{"style":550},[57,9598],{"className":9599,"style":154},[153],[57,9601,9603],{"className":9602},[158,159,160,161],[57,9604,280],{"className":9605},[123,161],[57,9607,169],{"className":9608},[168],[57,9610,9612],{"className":9611},[141],[57,9613,9615],{"className":9614,"style":176},[145],[57,9616],{}," 在 ",[57,9619,9621,9648],{"className":9620},[60],[57,9622,9624],{"className":9623},[64],[66,9625,9626],{"xmlns":68},[70,9627,9628,9646],{},[73,9629,9630,9636,9638,9644],{},[80,9631,9632,9634],{},[83,9633,2354],{},[278,9635,290],{},[76,9637,283],{"separator":98},[80,9639,9640,9642],{},[83,9641,2354],{},[278,9643,299],{},[76,9645,506],{},[90,9647,5830],{"encoding":92},[57,9649,9651],{"className":9650,"ariaHidden":98},[97],[57,9652,9654,9657,9697,9700,9703,9743,9746],{"className":9653},[102],[57,9655],{"className":9656,"style":320},[106],[57,9658,9660,9663],{"className":9659},[123],[57,9661,2354],{"className":9662,"style":2387},[123,127],[57,9664,9666],{"className":9665},[132],[57,9667,9669,9689],{"className":9668},[136,137],[57,9670,9672,9686],{"className":9671},[141],[57,9673,9675],{"className":9674,"style":339},[145],[57,9676,9677,9680],{"style":2402},[57,9678],{"className":9679,"style":154},[153],[57,9681,9683],{"className":9682},[158,159,160,161],[57,9684,290],{"className":9685},[123,161],[57,9687,169],{"className":9688},[168],[57,9690,9692],{"className":9691},[141],[57,9693,9695],{"className":9694,"style":176},[145],[57,9696],{},[57,9698,283],{"className":9699},[365],[57,9701],{"className":9702,"style":119},[118],[57,9704,9706,9709],{"className":9705},[123],[57,9707,2354],{"className":9708,"style":2387},[123,127],[57,9710,9712],{"className":9711},[132],[57,9713,9715,9735],{"className":9714},[136,137],[57,9716,9718,9732],{"className":9717},[141],[57,9719,9721],{"className":9720,"style":339},[145],[57,9722,9723,9726],{"style":2402},[57,9724],{"className":9725,"style":154},[153],[57,9727,9729],{"className":9728},[158,159,160,161],[57,9730,299],{"className":9731},[123,161],[57,9733,169],{"className":9734},[168],[57,9736,9738],{"className":9737},[141],[57,9739,9741],{"className":9740,"style":176},[145],[57,9742],{},[57,9744],{"className":9745,"style":119},[118],[57,9747,506],{"className":9748},[672]," 里都是 0。",[21,9751,9752,9753,5141,9822,10156,10157,10289,10290,10414,10415,10484,10485,10554,10555],{},"考察上面的等式两边同时模 ",[57,9754,9756,9773],{"className":9755},[60],[57,9757,9759],{"className":9758},[64],[66,9760,9761],{"xmlns":68},[70,9762,9763,9771],{},[73,9764,9765],{},[80,9766,9767,9769],{},[83,9768,2354],{},[278,9770,280],{},[90,9772,5005],{"encoding":92},[57,9774,9776],{"className":9775,"ariaHidden":98},[97],[57,9777,9779,9782],{"className":9778},[102],[57,9780],{"className":9781,"style":219},[106],[57,9783,9785,9788],{"className":9784},[123],[57,9786,2354],{"className":9787,"style":2387},[123,127],[57,9789,9791],{"className":9790},[132],[57,9792,9794,9814],{"className":9793},[136,137],[57,9795,9797,9811],{"className":9796},[141],[57,9798,9800],{"className":9799,"style":339},[145],[57,9801,9802,9805],{"style":2402},[57,9803],{"className":9804,"style":154},[153],[57,9806,9808],{"className":9807},[158,159,160,161],[57,9809,280],{"className":9810},[123,161],[57,9812,169],{"className":9813},[168],[57,9815,9817],{"className":9816},[141],[57,9818,9820],{"className":9819,"style":176},[145],[57,9821],{},[57,9823,9825,9891],{"className":9824},[60],[57,9826,9828],{"className":9827},[64],[66,9829,9830],{"xmlns":68},[70,9831,9832,9888],{},[73,9833,9834,9836,9838,9844,9846,9852,9854,9856,9858,9864,9866,9868,9870,9878,9880,9886],{},[278,9835,280],{},[76,9837,3926],{},[80,9839,9840,9842],{},[83,9841,3941],{},[278,9843,280],{},[76,9845,1008],{},[80,9847,9848,9850],{},[83,9849,3941],{},[278,9851,290],{},[76,9853,1008],{},[76,9855,4483],{},[76,9857,1008],{},[80,9859,9860,9862],{},[83,9861,3941],{},[83,9863,2206],{},[118,9865],{},[118,9867],{"width":3934},[76,9869,478],{"stretchy":477},[73,9871,9872,9874,9876],{},[83,9873,3941],{"mathvariant":302},[83,9875,3944],{"mathvariant":302},[83,9877,3947],{"mathvariant":302},[118,9879],{"width":3950},[80,9881,9882,9884],{},[83,9883,2354],{},[278,9885,280],{},[76,9887,513],{"stretchy":477},[90,9889,9890],{"encoding":92},"1 \\equiv m_1 + m_2 + \\dots + m_n \\pmod{I_1}",[57,9892,9894,9912,9967,10022,10040,10092],{"className":9893,"ariaHidden":98},[97],[57,9895,9897,9900,9903,9906,9909],{"className":9896},[102],[57,9898],{"className":9899,"style":3167},[106],[57,9901,280],{"className":9902},[123],[57,9904],{"className":9905,"style":1281},[118],[57,9907,3926],{"className":9908},[1285],[57,9910],{"className":9911,"style":1281},[118],[57,9913,9915,9918,9958,9961,9964],{"className":9914},[102],[57,9916],{"className":9917,"style":9027},[106],[57,9919,9921,9924],{"className":9920},[123],[57,9922,3941],{"className":9923},[123,127],[57,9925,9927],{"className":9926},[132],[57,9928,9930,9950],{"className":9929},[136,137],[57,9931,9933,9947],{"className":9932},[141],[57,9934,9936],{"className":9935,"style":339},[145],[57,9937,9938,9941],{"style":550},[57,9939],{"className":9940,"style":154},[153],[57,9942,9944],{"className":9943},[158,159,160,161],[57,9945,280],{"className":9946},[123,161],[57,9948,169],{"className":9949},[168],[57,9951,9953],{"className":9952},[141],[57,9954,9956],{"className":9955,"style":176},[145],[57,9957],{},[57,9959],{"className":9960,"style":1172},[118],[57,9962,1008],{"className":9963},[1176],[57,9965],{"className":9966,"style":1172},[118],[57,9968,9970,9973,10013,10016,10019],{"className":9969},[102],[57,9971],{"className":9972,"style":9027},[106],[57,9974,9976,9979],{"className":9975},[123],[57,9977,3941],{"className":9978},[123,127],[57,9980,9982],{"className":9981},[132],[57,9983,9985,10005],{"className":9984},[136,137],[57,9986,9988,10002],{"className":9987},[141],[57,9989,9991],{"className":9990,"style":339},[145],[57,9992,9993,9996],{"style":550},[57,9994],{"className":9995,"style":154},[153],[57,9997,9999],{"className":9998},[158,159,160,161],[57,10000,290],{"className":10001},[123,161],[57,10003,169],{"className":10004},[168],[57,10006,10008],{"className":10007},[141],[57,10009,10011],{"className":10010,"style":176},[145],[57,10012],{},[57,10014],{"className":10015,"style":1172},[118],[57,10017,1008],{"className":10018},[1176],[57,10020],{"className":10021,"style":1172},[118],[57,10023,10025,10028,10031,10034,10037],{"className":10024},[102],[57,10026],{"className":10027,"style":3129},[106],[57,10029,4483],{"className":10030},[672],[57,10032],{"className":10033,"style":1172},[118],[57,10035,1008],{"className":10036},[1176],[57,10038],{"className":10039,"style":1172},[118],[57,10041,10043,10046,10086,10089],{"className":10042},[102],[57,10044],{"className":10045,"style":711},[106],[57,10047,10049,10052],{"className":10048},[123],[57,10050,3941],{"className":10051},[123,127],[57,10053,10055],{"className":10054},[132],[57,10056,10058,10078],{"className":10057},[136,137],[57,10059,10061,10075],{"className":10060},[141],[57,10062,10064],{"className":10063,"style":2295},[145],[57,10065,10066,10069],{"style":550},[57,10067],{"className":10068,"style":154},[153],[57,10070,10072],{"className":10071},[158,159,160,161],[57,10073,2206],{"className":10074},[123,127,161],[57,10076,169],{"className":10077},[168],[57,10079,10081],{"className":10080},[141],[57,10082,10084],{"className":10083,"style":176},[145],[57,10085],{},[57,10087],{"className":10088},[118,3996],[57,10090],{"className":10091,"style":4000},[118],[57,10093,10095,10098,10101,10110,10113,10153],{"className":10094},[102],[57,10096],{"className":10097,"style":107},[106],[57,10099,478],{"className":10100},[529],[57,10102,10104],{"className":10103},[123],[57,10105,10107],{"className":10106},[123],[57,10108,4020],{"className":10109},[123,4019],[57,10111],{"className":10112,"style":4024},[118],[57,10114,10116,10119],{"className":10115},[123],[57,10117,2354],{"className":10118,"style":2387},[123,127],[57,10120,10122],{"className":10121},[132],[57,10123,10125,10145],{"className":10124},[136,137],[57,10126,10128,10142],{"className":10127},[141],[57,10129,10131],{"className":10130,"style":339},[145],[57,10132,10133,10136],{"style":2402},[57,10134],{"className":10135,"style":154},[153],[57,10137,10139],{"className":10138},[158,159,160,161],[57,10140,280],{"className":10141},[123,161],[57,10143,169],{"className":10144},[168],[57,10146,10148],{"className":10147},[141],[57,10149,10151],{"className":10150,"style":176},[145],[57,10152],{},[57,10154,513],{"className":10155},[679],"\n因为 ",[57,10158,10160,10188],{"className":10159},[60],[57,10161,10163],{"className":10162},[64],[66,10164,10165],{"xmlns":68},[70,10166,10167,10185],{},[73,10168,10169,10175,10177,10183],{},[80,10170,10171,10173],{},[83,10172,3941],{},[278,10174,290],{},[76,10176,283],{"separator":98},[80,10178,10179,10181],{},[83,10180,3941],{},[278,10182,299],{},[76,10184,506],{},[90,10186,10187],{"encoding":92},"m_2, m_3 \\dots",[57,10189,10191],{"className":10190,"ariaHidden":98},[97],[57,10192,10194,10197,10237,10240,10243,10283,10286],{"className":10193},[102],[57,10195],{"className":10196,"style":3148},[106],[57,10198,10200,10203],{"className":10199},[123],[57,10201,3941],{"className":10202},[123,127],[57,10204,10206],{"className":10205},[132],[57,10207,10209,10229],{"className":10208},[136,137],[57,10210,10212,10226],{"className":10211},[141],[57,10213,10215],{"className":10214,"style":339},[145],[57,10216,10217,10220],{"style":550},[57,10218],{"className":10219,"style":154},[153],[57,10221,10223],{"className":10222},[158,159,160,161],[57,10224,290],{"className":10225},[123,161],[57,10227,169],{"className":10228},[168],[57,10230,10232],{"className":10231},[141],[57,10233,10235],{"className":10234,"style":176},[145],[57,10236],{},[57,10238,283],{"className":10239},[365],[57,10241],{"className":10242,"style":119},[118],[57,10244,10246,10249],{"className":10245},[123],[57,10247,3941],{"className":10248},[123,127],[57,10250,10252],{"className":10251},[132],[57,10253,10255,10275],{"className":10254},[136,137],[57,10256,10258,10272],{"className":10257},[141],[57,10259,10261],{"className":10260,"style":339},[145],[57,10262,10263,10266],{"style":550},[57,10264],{"className":10265,"style":154},[153],[57,10267,10269],{"className":10268},[158,159,160,161],[57,10270,299],{"className":10271},[123,161],[57,10273,169],{"className":10274},[168],[57,10276,10278],{"className":10277},[141],[57,10279,10281],{"className":10280,"style":176},[145],[57,10282],{},[57,10284],{"className":10285,"style":119},[118],[57,10287,506],{"className":10288},[672]," 都在 ",[57,10291,10293,10319],{"className":10292},[60],[57,10294,10296],{"className":10295},[64],[66,10297,10298],{"xmlns":68},[70,10299,10300,10316],{},[73,10301,10302,10308,10310],{},[80,10303,10304,10306],{},[83,10305,5084],{},[278,10307,290],{},[76,10309,283],{"separator":98},[80,10311,10312,10314],{},[83,10313,5084],{},[278,10315,299],{},[90,10317,10318],{"encoding":92},"M_2, M_3",[57,10320,10322],{"className":10321,"ariaHidden":98},[97],[57,10323,10325,10328,10368,10371,10374],{"className":10324},[102],[57,10326],{"className":10327,"style":320},[106],[57,10329,10331,10334],{"className":10330},[123],[57,10332,5084],{"className":10333,"style":5105},[123,127],[57,10335,10337],{"className":10336},[132],[57,10338,10340,10360],{"className":10339},[136,137],[57,10341,10343,10357],{"className":10342},[141],[57,10344,10346],{"className":10345,"style":339},[145],[57,10347,10348,10351],{"style":5120},[57,10349],{"className":10350,"style":154},[153],[57,10352,10354],{"className":10353},[158,159,160,161],[57,10355,290],{"className":10356},[123,161],[57,10358,169],{"className":10359},[168],[57,10361,10363],{"className":10362},[141],[57,10364,10366],{"className":10365,"style":176},[145],[57,10367],{},[57,10369,283],{"className":10370},[365],[57,10372],{"className":10373,"style":119},[118],[57,10375,10377,10380],{"className":10376},[123],[57,10378,5084],{"className":10379,"style":5105},[123,127],[57,10381,10383],{"className":10382},[132],[57,10384,10386,10406],{"className":10385},[136,137],[57,10387,10389,10403],{"className":10388},[141],[57,10390,10392],{"className":10391,"style":339},[145],[57,10393,10394,10397],{"style":5120},[57,10395],{"className":10396,"style":154},[153],[57,10398,10400],{"className":10399},[158,159,160,161],[57,10401,299],{"className":10402},[123,161],[57,10404,169],{"className":10405},[168],[57,10407,10409],{"className":10408},[141],[57,10410,10412],{"className":10411,"style":176},[145],[57,10413],{}," 里（都包含 ",[57,10416,10418,10435],{"className":10417},[60],[57,10419,10421],{"className":10420},[64],[66,10422,10423],{"xmlns":68},[70,10424,10425,10433],{},[73,10426,10427],{},[80,10428,10429,10431],{},[83,10430,2354],{},[278,10432,280],{},[90,10434,5005],{"encoding":92},[57,10436,10438],{"className":10437,"ariaHidden":98},[97],[57,10439,10441,10444],{"className":10440},[102],[57,10442],{"className":10443,"style":219},[106],[57,10445,10447,10450],{"className":10446},[123],[57,10448,2354],{"className":10449,"style":2387},[123,127],[57,10451,10453],{"className":10452},[132],[57,10454,10456,10476],{"className":10455},[136,137],[57,10457,10459,10473],{"className":10458},[141],[57,10460,10462],{"className":10461,"style":339},[145],[57,10463,10464,10467],{"style":2402},[57,10465],{"className":10466,"style":154},[153],[57,10468,10470],{"className":10469},[158,159,160,161],[57,10471,280],{"className":10472},[123,161],[57,10474,169],{"className":10475},[168],[57,10477,10479],{"className":10478},[141],[57,10480,10482],{"className":10481,"style":176},[145],[57,10483],{}," 这个因子），所以它们模 ",[57,10486,10488,10505],{"className":10487},[60],[57,10489,10491],{"className":10490},[64],[66,10492,10493],{"xmlns":68},[70,10494,10495,10503],{},[73,10496,10497],{},[80,10498,10499,10501],{},[83,10500,2354],{},[278,10502,280],{},[90,10504,5005],{"encoding":92},[57,10506,10508],{"className":10507,"ariaHidden":98},[97],[57,10509,10511,10514],{"className":10510},[102],[57,10512],{"className":10513,"style":219},[106],[57,10515,10517,10520],{"className":10516},[123],[57,10518,2354],{"className":10519,"style":2387},[123,127],[57,10521,10523],{"className":10522},[132],[57,10524,10526,10546],{"className":10525},[136,137],[57,10527,10529,10543],{"className":10528},[141],[57,10530,10532],{"className":10531,"style":339},[145],[57,10533,10534,10537],{"style":2402},[57,10535],{"className":10536,"style":154},[153],[57,10538,10540],{"className":10539},[158,159,160,161],[57,10541,280],{"className":10542},[123,161],[57,10544,169],{"className":10545},[168],[57,10547,10549],{"className":10548},[141],[57,10550,10552],{"className":10551,"style":176},[145],[57,10553],{}," 都是 0。\n",[57,10556,10558,10658],{"className":10557},[60],[57,10559,10561],{"className":10560},[64],[66,10562,10563],{"xmlns":68},[70,10564,10565,10655],{},[73,10566,10567,10569,10571,10577,10579,10581,10583,10585,10587,10589,10591,10593,10595,10603,10605,10611,10613,10616,10619,10621,10627,10629,10631,10633,10635,10637,10645,10647,10653],{},[278,10568,280],{},[76,10570,3926],{},[80,10572,10573,10575],{},[83,10574,3941],{},[278,10576,280],{},[76,10578,1008],{},[278,10580,3929],{},[76,10582,1008],{},[76,10584,4483],{},[76,10586,1008],{},[278,10588,3929],{},[118,10590],{},[118,10592],{"width":3934},[76,10594,478],{"stretchy":477},[73,10596,10597,10599,10601],{},[83,10598,3941],{"mathvariant":302},[83,10600,3944],{"mathvariant":302},[83,10602,3947],{"mathvariant":302},[118,10604],{"width":3950},[80,10606,10607,10609],{},[83,10608,2354],{},[278,10610,280],{},[76,10612,513],{"stretchy":477},[508,10614,10615],{},"  ",[76,10617,10618],{},"⟹",[508,10620,10615],{},[80,10622,10623,10625],{},[83,10624,3941],{},[278,10626,280],{},[76,10628,3926],{},[278,10630,280],{},[118,10632],{},[118,10634],{"width":3934},[76,10636,478],{"stretchy":477},[73,10638,10639,10641,10643],{},[83,10640,3941],{"mathvariant":302},[83,10642,3944],{"mathvariant":302},[83,10644,3947],{"mathvariant":302},[118,10646],{"width":3950},[80,10648,10649,10651],{},[83,10650,2354],{},[278,10652,280],{},[76,10654,513],{"stretchy":477},[90,10656,10657],{"encoding":92},"1 \\equiv m_1 + 0 + \\dots + 0 \\pmod{I_1} \\implies m_1 \\equiv 1 \\pmod{I_1}",[57,10659,10661,10679,10734,10753,10771,10786,10865,10921,10936],{"className":10660,"ariaHidden":98},[97],[57,10662,10664,10667,10670,10673,10676],{"className":10663},[102],[57,10665],{"className":10666,"style":3167},[106],[57,10668,280],{"className":10669},[123],[57,10671],{"className":10672,"style":1281},[118],[57,10674,3926],{"className":10675},[1285],[57,10677],{"className":10678,"style":1281},[118],[57,10680,10682,10685,10725,10728,10731],{"className":10681},[102],[57,10683],{"className":10684,"style":9027},[106],[57,10686,10688,10691],{"className":10687},[123],[57,10689,3941],{"className":10690},[123,127],[57,10692,10694],{"className":10693},[132],[57,10695,10697,10717],{"className":10696},[136,137],[57,10698,10700,10714],{"className":10699},[141],[57,10701,10703],{"className":10702,"style":339},[145],[57,10704,10705,10708],{"style":550},[57,10706],{"className":10707,"style":154},[153],[57,10709,10711],{"className":10710},[158,159,160,161],[57,10712,280],{"className":10713},[123,161],[57,10715,169],{"className":10716},[168],[57,10718,10720],{"className":10719},[141],[57,10721,10723],{"className":10722,"style":176},[145],[57,10724],{},[57,10726],{"className":10727,"style":1172},[118],[57,10729,1008],{"className":10730},[1176],[57,10732],{"className":10733,"style":1172},[118],[57,10735,10737,10741,10744,10747,10750],{"className":10736},[102],[57,10738],{"className":10739,"style":10740},[106],"height:0.7278em;vertical-align:-0.0833em;",[57,10742,3929],{"className":10743},[123],[57,10745],{"className":10746,"style":1172},[118],[57,10748,1008],{"className":10749},[1176],[57,10751],{"className":10752,"style":1172},[118],[57,10754,10756,10759,10762,10765,10768],{"className":10755},[102],[57,10757],{"className":10758,"style":3129},[106],[57,10760,4483],{"className":10761},[672],[57,10763],{"className":10764,"style":1172},[118],[57,10766,1008],{"className":10767},[1176],[57,10769],{"className":10770,"style":1172},[118],[57,10772,10774,10777,10780,10783],{"className":10773},[102],[57,10775],{"className":10776,"style":3167},[106],[57,10778,3929],{"className":10779},[123],[57,10781],{"className":10782},[118,3996],[57,10784],{"className":10785,"style":4000},[118],[57,10787,10789,10792,10795,10804,10807,10847,10850,10853,10856,10859,10862],{"className":10788},[102],[57,10790],{"className":10791,"style":107},[106],[57,10793,478],{"className":10794},[529],[57,10796,10798],{"className":10797},[123],[57,10799,10801],{"className":10800},[123],[57,10802,4020],{"className":10803},[123,4019],[57,10805],{"className":10806,"style":4024},[118],[57,10808,10810,10813],{"className":10809},[123],[57,10811,2354],{"className":10812,"style":2387},[123,127],[57,10814,10816],{"className":10815},[132],[57,10817,10819,10839],{"className":10818},[136,137],[57,10820,10822,10836],{"className":10821},[141],[57,10823,10825],{"className":10824,"style":339},[145],[57,10826,10827,10830],{"style":2402},[57,10828],{"className":10829,"style":154},[153],[57,10831,10833],{"className":10832},[158,159,160,161],[57,10834,280],{"className":10835},[123,161],[57,10837,169],{"className":10838},[168],[57,10840,10842],{"className":10841},[141],[57,10843,10845],{"className":10844,"style":176},[145],[57,10846],{},[57,10848,513],{"className":10849},[679],[57,10851],{"className":10852,"style":1281},[118],[57,10854],{"className":10855,"style":1281},[118],[57,10857,10618],{"className":10858},[1285],[57,10860],{"className":10861,"style":1281},[118],[57,10863],{"className":10864,"style":1281},[118],[57,10866,10868,10872,10912,10915,10918],{"className":10867},[102],[57,10869],{"className":10870,"style":10871},[106],"height:0.6138em;vertical-align:-0.15em;",[57,10873,10875,10878],{"className":10874},[123],[57,10876,3941],{"className":10877},[123,127],[57,10879,10881],{"className":10880},[132],[57,10882,10884,10904],{"className":10883},[136,137],[57,10885,10887,10901],{"className":10886},[141],[57,10888,10890],{"className":10889,"style":339},[145],[57,10891,10892,10895],{"style":550},[57,10893],{"className":10894,"style":154},[153],[57,10896,10898],{"className":10897},[158,159,160,161],[57,10899,280],{"className":10900},[123,161],[57,10902,169],{"className":10903},[168],[57,10905,10907],{"className":10906},[141],[57,10908,10910],{"className":10909,"style":176},[145],[57,10911],{},[57,10913],{"className":10914,"style":1281},[118],[57,10916,3926],{"className":10917},[1285],[57,10919],{"className":10920,"style":1281},[118],[57,10922,10924,10927,10930,10933],{"className":10923},[102],[57,10925],{"className":10926,"style":3167},[106],[57,10928,280],{"className":10929},[123],[57,10931],{"className":10932},[118,3996],[57,10934],{"className":10935,"style":4000},[118],[57,10937,10939,10942,10945,10954,10957,10997],{"className":10938},[102],[57,10940],{"className":10941,"style":107},[106],[57,10943,478],{"className":10944},[529],[57,10946,10948],{"className":10947},[123],[57,10949,10951],{"className":10950},[123],[57,10952,4020],{"className":10953},[123,4019],[57,10955],{"className":10956,"style":4024},[118],[57,10958,10960,10963],{"className":10959},[123],[57,10961,2354],{"className":10962,"style":2387},[123,127],[57,10964,10966],{"className":10965},[132],[57,10967,10969,10989],{"className":10968},[136,137],[57,10970,10972,10986],{"className":10971},[141],[57,10973,10975],{"className":10974,"style":339},[145],[57,10976,10977,10980],{"style":2402},[57,10978],{"className":10979,"style":154},[153],[57,10981,10983],{"className":10982},[158,159,160,161],[57,10984,280],{"className":10985},[123,161],[57,10987,169],{"className":10988},[168],[57,10990,10992],{"className":10991},[141],[57,10993,10995],{"className":10994,"style":176},[145],[57,10996],{},[57,10998,513],{"className":10999},[679],[6679,11001],{},[24,11003,11004,11005,11075],{},"总结 ",[57,11006,11008,11026],{"className":11007},[60],[57,11009,11011],{"className":11010},[64],[66,11012,11013],{"xmlns":68},[70,11014,11015,11023],{},[73,11016,11017],{},[80,11018,11019,11021],{},[83,11020,3941],{},[83,11022,88],{},[90,11024,11025],{"encoding":92},"m_i",[57,11027,11029],{"className":11028,"ariaHidden":98},[97],[57,11030,11032,11035],{"className":11031},[102],[57,11033],{"className":11034,"style":711},[106],[57,11036,11038,11041],{"className":11037},[123],[57,11039,3941],{"className":11040},[123,127],[57,11042,11044],{"className":11043},[132],[57,11045,11047,11067],{"className":11046},[136,137],[57,11048,11050,11064],{"className":11049},[141],[57,11051,11053],{"className":11052,"style":146},[145],[57,11054,11055,11058],{"style":550},[57,11056],{"className":11057,"style":154},[153],[57,11059,11061],{"className":11060},[158,159,160,161],[57,11062,88],{"className":11063},[123,127,161],[57,11065,169],{"className":11066},[168],[57,11068,11070],{"className":11069},[141],[57,11071,11073],{"className":11072,"style":176},[145],[57,11074],{}," 的性质",[184,11077,11078,11180],{},[21,11079,11080,11149,11150,11179],{},[57,11081,11083,11100],{"className":11082},[60],[57,11084,11086],{"className":11085},[64],[66,11087,11088],{"xmlns":68},[70,11089,11090,11098],{},[73,11091,11092],{},[80,11093,11094,11096],{},[83,11095,3941],{},[83,11097,88],{},[90,11099,11025],{"encoding":92},[57,11101,11103],{"className":11102,"ariaHidden":98},[97],[57,11104,11106,11109],{"className":11105},[102],[57,11107],{"className":11108,"style":711},[106],[57,11110,11112,11115],{"className":11111},[123],[57,11113,3941],{"className":11114},[123,127],[57,11116,11118],{"className":11117},[132],[57,11119,11121,11141],{"className":11120},[136,137],[57,11122,11124,11138],{"className":11123},[141],[57,11125,11127],{"className":11126,"style":146},[145],[57,11128,11129,11132],{"style":550},[57,11130],{"className":11131,"style":154},[153],[57,11133,11135],{"className":11134},[158,159,160,161],[57,11136,88],{"className":11137},[123,127,161],[57,11139,169],{"className":11140},[168],[57,11142,11144],{"className":11143},[141],[57,11145,11147],{"className":11146,"style":176},[145],[57,11148],{}," 在第 ",[57,11151,11153,11166],{"className":11152},[60],[57,11154,11156],{"className":11155},[64],[66,11157,11158],{"xmlns":68},[70,11159,11160,11164],{},[73,11161,11162],{},[83,11163,88],{},[90,11165,88],{"encoding":92},[57,11167,11169],{"className":11168,"ariaHidden":98},[97],[57,11170,11172,11176],{"className":11171},[102],[57,11173],{"className":11174,"style":11175},[106],"height:0.6595em;",[57,11177,88],{"className":11178},[123,127]," 个位置看是 1。",[21,11181,11182,11251],{},[57,11183,11185,11202],{"className":11184},[60],[57,11186,11188],{"className":11187},[64],[66,11189,11190],{"xmlns":68},[70,11191,11192,11200],{},[73,11193,11194],{},[80,11195,11196,11198],{},[83,11197,3941],{},[83,11199,88],{},[90,11201,11025],{"encoding":92},[57,11203,11205],{"className":11204,"ariaHidden":98},[97],[57,11206,11208,11211],{"className":11207},[102],[57,11209],{"className":11210,"style":711},[106],[57,11212,11214,11217],{"className":11213},[123],[57,11215,3941],{"className":11216},[123,127],[57,11218,11220],{"className":11219},[132],[57,11221,11223,11243],{"className":11222},[136,137],[57,11224,11226,11240],{"className":11225},[141],[57,11227,11229],{"className":11228,"style":146},[145],[57,11230,11231,11234],{"style":550},[57,11232],{"className":11233,"style":154},[153],[57,11235,11237],{"className":11236},[158,159,160,161],[57,11238,88],{"className":11239},[123,127,161],[57,11241,169],{"className":11242},[168],[57,11244,11246],{"className":11245},[141],[57,11247,11249],{"className":11248,"style":176},[145],[57,11250],{}," 在别的位置看是 0。",[21,11253,11254,11257,11258,11462,11463,11841,11843,11844,11888,11889,12039,12040,12068,12069,12139,12140,13247,13249,13252],{},[24,11255,11256],{},"最后合成解：","\n现在对于任意的目标列表 ",[57,11259,11261,11303],{"className":11260},[60],[57,11262,11264],{"className":11263},[64],[66,11265,11266],{"xmlns":68},[70,11267,11268,11300],{},[73,11269,11270,11272,11278,11280,11286,11288,11290,11292,11298],{},[76,11271,478],{"stretchy":477},[80,11273,11274,11276],{},[83,11275,3000],{},[278,11277,280],{},[76,11279,283],{"separator":98},[80,11281,11282,11284],{},[83,11283,3000],{},[278,11285,290],{},[76,11287,283],{"separator":98},[76,11289,506],{},[76,11291,283],{"separator":98},[80,11293,11294,11296],{},[83,11295,3000],{},[83,11297,2206],{},[76,11299,513],{"stretchy":477},[90,11301,11302],{"encoding":92},"(x_1, x_2, \\dots, x_n)",[57,11304,11306],{"className":11305,"ariaHidden":98},[97],[57,11307,11309,11312,11315,11355,11358,11361,11401,11404,11407,11410,11413,11416,11419,11459],{"className":11308},[102],[57,11310],{"className":11311,"style":107},[106],[57,11313,478],{"className":11314},[529],[57,11316,11318,11321],{"className":11317},[123],[57,11319,3000],{"className":11320},[123,127],[57,11322,11324],{"className":11323},[132],[57,11325,11327,11347],{"className":11326},[136,137],[57,11328,11330,11344],{"className":11329},[141],[57,11331,11333],{"className":11332,"style":339},[145],[57,11334,11335,11338],{"style":550},[57,11336],{"className":11337,"style":154},[153],[57,11339,11341],{"className":11340},[158,159,160,161],[57,11342,280],{"className":11343},[123,161],[57,11345,169],{"className":11346},[168],[57,11348,11350],{"className":11349},[141],[57,11351,11353],{"className":11352,"style":176},[145],[57,11354],{},[57,11356,283],{"className":11357},[365],[57,11359],{"className":11360,"style":119},[118],[57,11362,11364,11367],{"className":11363},[123],[57,11365,3000],{"className":11366},[123,127],[57,11368,11370],{"className":11369},[132],[57,11371,11373,11393],{"className":11372},[136,137],[57,11374,11376,11390],{"className":11375},[141],[57,11377,11379],{"className":11378,"style":339},[145],[57,11380,11381,11384],{"style":550},[57,11382],{"className":11383,"style":154},[153],[57,11385,11387],{"className":11386},[158,159,160,161],[57,11388,290],{"className":11389},[123,161],[57,11391,169],{"className":11392},[168],[57,11394,11396],{"className":11395},[141],[57,11397,11399],{"className":11398,"style":176},[145],[57,11400],{},[57,11402,283],{"className":11403},[365],[57,11405],{"className":11406,"style":119},[118],[57,11408,506],{"className":11409},[672],[57,11411],{"className":11412,"style":119},[118],[57,11414,283],{"className":11415},[365],[57,11417],{"className":11418,"style":119},[118],[57,11420,11422,11425],{"className":11421},[123],[57,11423,3000],{"className":11424},[123,127],[57,11426,11428],{"className":11427},[132],[57,11429,11431,11451],{"className":11430},[136,137],[57,11432,11434,11448],{"className":11433},[141],[57,11435,11437],{"className":11436,"style":2295},[145],[57,11438,11439,11442],{"style":550},[57,11440],{"className":11441,"style":154},[153],[57,11443,11445],{"className":11444},[158,159,160,161],[57,11446,2206],{"className":11447},[123,127,161],[57,11449,169],{"className":11450},[168],[57,11452,11454],{"className":11453},[141],[57,11455,11457],{"className":11456,"style":176},[145],[57,11458],{},[57,11460,513],{"className":11461},[679],"，我们令：\n",[57,11464,11466,11526],{"className":11465},[60],[57,11467,11469],{"className":11468},[64],[66,11470,11471],{"xmlns":68},[70,11472,11473,11523],{},[73,11474,11475,11477,11479,11485,11491,11493,11499,11505,11507,11509,11511,11517],{},[83,11476,3000],{},[76,11478,1030],{},[80,11480,11481,11483],{},[83,11482,3941],{},[278,11484,280],{},[80,11486,11487,11489],{},[83,11488,3000],{},[278,11490,280],{},[76,11492,1008],{},[80,11494,11495,11497],{},[83,11496,3941],{},[278,11498,290],{},[80,11500,11501,11503],{},[83,11502,3000],{},[278,11504,290],{},[76,11506,1008],{},[76,11508,4483],{},[76,11510,1008],{},[80,11512,11513,11515],{},[83,11514,3941],{},[83,11516,2206],{},[80,11518,11519,11521],{},[83,11520,3000],{},[83,11522,2206],{},[90,11524,11525],{"encoding":92},"x = m_1 x_1 + m_2 x_2 + \\dots + m_n x_n",[57,11527,11529,11547,11642,11737,11755],{"className":11528,"ariaHidden":98},[97],[57,11530,11532,11535,11538,11541,11544],{"className":11531},[102],[57,11533],{"className":11534,"style":3686},[106],[57,11536,3000],{"className":11537},[123,127],[57,11539],{"className":11540,"style":1281},[118],[57,11542,1030],{"className":11543},[1285],[57,11545],{"className":11546,"style":1281},[118],[57,11548,11550,11553,11593,11633,11636,11639],{"className":11549},[102],[57,11551],{"className":11552,"style":9027},[106],[57,11554,11556,11559],{"className":11555},[123],[57,11557,3941],{"className":11558},[123,127],[57,11560,11562],{"className":11561},[132],[57,11563,11565,11585],{"className":11564},[136,137],[57,11566,11568,11582],{"className":11567},[141],[57,11569,11571],{"className":11570,"style":339},[145],[57,11572,11573,11576],{"style":550},[57,11574],{"className":11575,"style":154},[153],[57,11577,11579],{"className":11578},[158,159,160,161],[57,11580,280],{"className":11581},[123,161],[57,11583,169],{"className":11584},[168],[57,11586,11588],{"className":11587},[141],[57,11589,11591],{"className":11590,"style":176},[145],[57,11592],{},[57,11594,11596,11599],{"className":11595},[123],[57,11597,3000],{"className":11598},[123,127],[57,11600,11602],{"className":11601},[132],[57,11603,11605,11625],{"className":11604},[136,137],[57,11606,11608,11622],{"className":11607},[141],[57,11609,11611],{"className":11610,"style":339},[145],[57,11612,11613,11616],{"style":550},[57,11614],{"className":11615,"style":154},[153],[57,11617,11619],{"className":11618},[158,159,160,161],[57,11620,280],{"className":11621},[123,161],[57,11623,169],{"className":11624},[168],[57,11626,11628],{"className":11627},[141],[57,11629,11631],{"className":11630,"style":176},[145],[57,11632],{},[57,11634],{"className":11635,"style":1172},[118],[57,11637,1008],{"className":11638},[1176],[57,11640],{"className":11641,"style":1172},[118],[57,11643,11645,11648,11688,11728,11731,11734],{"className":11644},[102],[57,11646],{"className":11647,"style":9027},[106],[57,11649,11651,11654],{"className":11650},[123],[57,11652,3941],{"className":11653},[123,127],[57,11655,11657],{"className":11656},[132],[57,11658,11660,11680],{"className":11659},[136,137],[57,11661,11663,11677],{"className":11662},[141],[57,11664,11666],{"className":11665,"style":339},[145],[57,11667,11668,11671],{"style":550},[57,11669],{"className":11670,"style":154},[153],[57,11672,11674],{"className":11673},[158,159,160,161],[57,11675,290],{"className":11676},[123,161],[57,11678,169],{"className":11679},[168],[57,11681,11683],{"className":11682},[141],[57,11684,11686],{"className":11685,"style":176},[145],[57,11687],{},[57,11689,11691,11694],{"className":11690},[123],[57,11692,3000],{"className":11693},[123,127],[57,11695,11697],{"className":11696},[132],[57,11698,11700,11720],{"className":11699},[136,137],[57,11701,11703,11717],{"className":11702},[141],[57,11704,11706],{"className":11705,"style":339},[145],[57,11707,11708,11711],{"style":550},[57,11709],{"className":11710,"style":154},[153],[57,11712,11714],{"className":11713},[158,159,160,161],[57,11715,290],{"className":11716},[123,161],[57,11718,169],{"className":11719},[168],[57,11721,11723],{"className":11722},[141],[57,11724,11726],{"className":11725,"style":176},[145],[57,11727],{},[57,11729],{"className":11730,"style":1172},[118],[57,11732,1008],{"className":11733},[1176],[57,11735],{"className":11736,"style":1172},[118],[57,11738,11740,11743,11746,11749,11752],{"className":11739},[102],[57,11741],{"className":11742,"style":3129},[106],[57,11744,4483],{"className":11745},[672],[57,11747],{"className":11748,"style":1172},[118],[57,11750,1008],{"className":11751},[1176],[57,11753],{"className":11754,"style":1172},[118],[57,11756,11758,11761,11801],{"className":11757},[102],[57,11759],{"className":11760,"style":711},[106],[57,11762,11764,11767],{"className":11763},[123],[57,11765,3941],{"className":11766},[123,127],[57,11768,11770],{"className":11769},[132],[57,11771,11773,11793],{"className":11772},[136,137],[57,11774,11776,11790],{"className":11775},[141],[57,11777,11779],{"className":11778,"style":2295},[145],[57,11780,11781,11784],{"style":550},[57,11782],{"className":11783,"style":154},[153],[57,11785,11787],{"className":11786},[158,159,160,161],[57,11788,2206],{"className":11789},[123,127,161],[57,11791,169],{"className":11792},[168],[57,11794,11796],{"className":11795},[141],[57,11797,11799],{"className":11798,"style":176},[145],[57,11800],{},[57,11802,11804,11807],{"className":11803},[123],[57,11805,3000],{"className":11806},[123,127],[57,11808,11810],{"className":11809},[132],[57,11811,11813,11833],{"className":11812},[136,137],[57,11814,11816,11830],{"className":11815},[141],[57,11817,11819],{"className":11818,"style":2295},[145],[57,11820,11821,11824],{"style":550},[57,11822],{"className":11823,"style":154},[153],[57,11825,11827],{"className":11826},[158,159,160,161],[57,11828,2206],{"className":11829},[123,127,161],[57,11831,169],{"className":11832},[168],[57,11834,11836],{"className":11835},[141],[57,11837,11839],{"className":11838,"style":176},[145],[57,11840],{},[6679,11842],{},"让我们验证一下 ",[57,11845,11847,11867],{"className":11846},[60],[57,11848,11850],{"className":11849},[64],[66,11851,11852],{"xmlns":68},[70,11853,11854,11864],{},[73,11855,11856,11858,11860,11862],{},[83,11857,3196],{},[76,11859,478],{"stretchy":477},[83,11861,3000],{},[76,11863,513],{"stretchy":477},[90,11865,11866],{"encoding":92},"\\phi(x)",[57,11868,11870],{"className":11869,"ariaHidden":98},[97],[57,11871,11873,11876,11879,11882,11885],{"className":11872},[102],[57,11874],{"className":11875,"style":107},[106],[57,11877,3196],{"className":11878},[123,127],[57,11880,478],{"className":11881},[529],[57,11883,3000],{"className":11884},[123,127],[57,11886,513],{"className":11887},[679]," 是不是等于 ",[57,11890,11892,11926],{"className":11891},[60],[57,11893,11895],{"className":11894},[64],[66,11896,11897],{"xmlns":68},[70,11898,11899,11923],{},[73,11900,11901,11903,11909,11911,11913,11915,11921],{},[76,11902,478],{"stretchy":477},[80,11904,11905,11907],{},[83,11906,3000],{},[278,11908,280],{},[76,11910,283],{"separator":98},[76,11912,506],{},[76,11914,283],{"separator":98},[80,11916,11917,11919],{},[83,11918,3000],{},[83,11920,2206],{},[76,11922,513],{"stretchy":477},[90,11924,11925],{"encoding":92},"(x_1, \\dots, x_n)",[57,11927,11929],{"className":11928,"ariaHidden":98},[97],[57,11930,11932,11935,11938,11978,11981,11984,11987,11990,11993,11996,12036],{"className":11931},[102],[57,11933],{"className":11934,"style":107},[106],[57,11936,478],{"className":11937},[529],[57,11939,11941,11944],{"className":11940},[123],[57,11942,3000],{"className":11943},[123,127],[57,11945,11947],{"className":11946},[132],[57,11948,11950,11970],{"className":11949},[136,137],[57,11951,11953,11967],{"className":11952},[141],[57,11954,11956],{"className":11955,"style":339},[145],[57,11957,11958,11961],{"style":550},[57,11959],{"className":11960,"style":154},[153],[57,11962,11964],{"className":11963},[158,159,160,161],[57,11965,280],{"className":11966},[123,161],[57,11968,169],{"className":11969},[168],[57,11971,11973],{"className":11972},[141],[57,11974,11976],{"className":11975,"style":176},[145],[57,11977],{},[57,11979,283],{"className":11980},[365],[57,11982],{"className":11983,"style":119},[118],[57,11985,506],{"className":11986},[672],[57,11988],{"className":11989,"style":119},[118],[57,11991,283],{"className":11992},[365],[57,11994],{"className":11995,"style":119},[118],[57,11997,11999,12002],{"className":11998},[123],[57,12000,3000],{"className":12001},[123,127],[57,12003,12005],{"className":12004},[132],[57,12006,12008,12028],{"className":12007},[136,137],[57,12009,12011,12025],{"className":12010},[141],[57,12012,12014],{"className":12013,"style":2295},[145],[57,12015,12016,12019],{"style":550},[57,12017],{"className":12018,"style":154},[153],[57,12020,12022],{"className":12021},[158,159,160,161],[57,12023,2206],{"className":12024},[123,127,161],[57,12026,169],{"className":12027},[168],[57,12029,12031],{"className":12030},[141],[57,12032,12034],{"className":12033,"style":176},[145],[57,12035],{},[57,12037,513],{"className":12038},[679],"。\n只看第 ",[57,12041,12043,12056],{"className":12042},[60],[57,12044,12046],{"className":12045},[64],[66,12047,12048],{"xmlns":68},[70,12049,12050,12054],{},[73,12051,12052],{},[83,12053,2545],{},[90,12055,2545],{"encoding":92},[57,12057,12059],{"className":12058,"ariaHidden":98},[97],[57,12060,12062,12065],{"className":12061},[102],[57,12063],{"className":12064,"style":7545},[106],[57,12066,2545],{"className":12067,"style":2643},[123,127]," 个分量（模 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&\\equiv m_1 x_1 + \\dots + m_k x_k + \\dots + m_n x_n \\pmod{I_k} \\\\\n&\\equiv 0 \\cdot x_1 + \\dots + 1 \\cdot x_k + \\dots + 0 \\cdot x_n \\pmod{I_k} \\\\\n&\\equiv x_k \\pmod{I_k}\n\\end{aligned}",[57,12381,12383],{"className":12382,"ariaHidden":98},[97],[57,12384,12386,12390],{"className":12385},[102],[57,12387],{"className":12388,"style":12389},[106],"height:4.2em;vertical-align:-1.85em;",[57,12391,12393],{"className":12392},[123],[57,12394,12396,12452],{"className":12395},[7571],[57,12397,12399],{"className":12398},[7753],[57,12400,12402,12443],{"className":12401},[136,137],[57,12403,12405,12440],{"className":12404},[141],[57,12406,12409,12422,12431],{"className":12407,"style":12408},[145],"height:2.35em;",[57,12410,12412,12416],{"style":12411},"top:-4.51em;",[57,12413],{"className":12414,"style":12415},[153],"height:3em;",[57,12417,12419],{"className":12418},[123],[57,12420,3000],{"className":12421},[123,127],[57,12423,12425,12428],{"style":12424},"top:-3.01em;",[57,12426],{"className":12427,"style":12415},[153],[57,12429],{"className":12430},[123],[57,12432,12434,12437],{"style":12433},"top:-1.51em;",[57,12435],{"className":12436,"style":12415},[153],[57,12438],{"className":12439},[123],[57,12441,169],{"className":12442},[168],[57,12444,12446],{"className":12445},[141],[57,12447,12450],{"className":12448,"style":12449},[145],"height:1.85em;",[57,12451],{},[57,12453,12455],{"className":12454},[7899],[57,12456,12458,13239],{"className":12457},[136,137],[57,12459,12461,13236],{"className":12460},[141],[57,12462,12464,12830,13112],{"className":12463,"style":12408},[145],[57,12465,12466,12469],{"style":12411},[57,12467],{"className":12468,"style":12415},[153],[57,12470,12472,12475,12478,12481,12484,12524,12564,12567,12570,12573,12576,12579,12582,12585,12625,12665,12668,12671,12674,12677,12680,12683,12686,12726,12766,12769,12772,12775,12784,12787,12827],{"className":12471},[123],[57,12473],{"className":12474},[123],[57,12476],{"className":12477,"style":1281},[118],[57,12479,3926],{"className":12480},[1285],[57,12482],{"className":12483,"style":1281},[118],[57,12485,12487,12490],{"className":12486},[123],[57,12488,3941],{"className":12489},[123,127],[57,12491,12493],{"className":12492},[132],[57,12494,12496,12516],{"className":12495},[136,137],[57,12497,12499,12513],{"className":12498},[141],[57,12500,12502],{"className":12501,"style":339},[145],[57,12503,12504,12507],{"style":550},[57,12505],{"className":12506,"style":154},[153],[57,12508,12510],{"className":12509},[158,159,160,161],[57,12511,280],{"className":12512},[123,161],[57,12514,169],{"className":12515},[168],[57,12517,12519],{"className":12518},[141],[57,12520,12522],{"className":12521,"style":176},[145],[57,12523],{},[57,12525,12527,12530],{"className":12526},[123],[57,12528,3000],{"className":12529},[123,127],[57,12531,12533],{"className":12532},[132],[57,12534,12536,12556],{"className":12535},[136,137],[57,12537,12539,12553],{"className":12538},[141],[57,12540,12542],{"className":12541,"style":339},[145],[57,12543,12544,12547],{"style":550},[57,12545],{"className":12546,"style":154},[153],[57,12548,12550],{"className":12549},[158,159,160,161],[57,12551,280],{"className":12552},[123,161],[57,12554,169],{"className":12555},[168],[57,12557,12559],{"className":12558},[141],[57,12560,12562],{"className":12561,"style":176},[145],[57,12563],{},[57,12565],{"className":12566,"style":1172},[118],[57,12568,1008],{"className":12569},[1176],[57,12571],{"className":12572,"style":1172},[118],[57,12574,4483],{"className":12575},[672],[57,12577],{"className":12578,"style":1172},[118],[57,12580,1008],{"className":12581},[1176],[57,12583],{"className":12584,"style":1172},[118],[57,12586,12588,12591],{"className":12587},[123],[57,12589,3941],{"className":12590},[123,127],[57,12592,12594],{"className":12593},[132],[57,12595,12597,12617],{"className":12596},[136,137],[57,12598,12600,12614],{"className":12599},[141],[57,12601,12603],{"className":12602,"style":2695},[145],[57,12604,12605,12608],{"style":550},[57,12606],{"className":12607,"style":154},[153],[57,12609,12611],{"className":12610},[158,159,160,161],[57,12612,2545],{"className":12613,"style":2643},[123,127,161],[57,12615,169],{"className":12616},[168],[57,12618,12620],{"className":12619},[141],[57,12621,12623],{"className":12622,"style":176},[145],[57,12624],{},[57,12626,12628,12631],{"className":12627},[123],[57,12629,3000],{"className":12630},[123,127],[57,12632,12634],{"className":12633},[132],[57,12635,12637,12657],{"className":12636},[136,137],[57,12638,12640,12654],{"className":12639},[141],[57,12641,12643],{"className":12642,"style":2695},[145],[57,12644,12645,12648],{"style":550},[57,12646],{"className":12647,"style":154},[153],[57,12649,12651],{"className":12650},[158,159,160,161],[57,12652,2545],{"className":12653,"style":2643},[123,127,161],[57,12655,169],{"className":12656},[168],[57,12658,12660],{"className":12659},[141],[57,12661,12663],{"className":12662,"style":176},[145],[57,12664],{},[57,12666],{"className":12667,"style":1172},[118],[57,12669,1008],{"className":12670},[1176],[57,12672],{"className":12673,"style":1172},[118],[57,12675,4483],{"className":126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每一个分量都完美对上了。所以是满射。",[35,13254],{},[38,13256,13258],{"id":13257},"第三阶段应用到整数推论-341","第三阶段：应用到整数（推论 3.4.1）",[14,13260,13261,13262,13290,13291,13324],{},"现在我们把抽象的“环 ",[57,13263,13265,13278],{"className":13264},[60],[57,13266,13268],{"className":13267},[64],[66,13269,13270],{"xmlns":68},[70,13271,13272,13276],{},[73,13273,13274],{},[83,13275,85],{},[90,13277,85],{"encoding":92},[57,13279,13281],{"className":13280,"ariaHidden":98},[97],[57,13282,13284,13287],{"className":13283},[102],[57,13285],{"className":13286,"style":2507},[106],[57,13288,85],{"className":13289,"style":128},[123,127],"”换成我们熟悉的“整数 ",[57,13292,13294,13310],{"className":13293},[60],[57,13295,13297],{"className":13296},[64],[66,13298,13299],{"xmlns":68},[70,13300,13301,13307],{},[73,13302,13303],{},[83,13304,13306],{"mathvariant":13305},"double-struck","Z",[90,13308,13309],{"encoding":92},"\\mathbb{Z}",[57,13311,13313],{"className":13312,"ariaHidden":98},[97],[57,13314,13316,13320],{"className":13315},[102],[57,13317],{"className":13318,"style":13319},[106],"height:0.6889em;",[57,13321,13306],{"className":13322},[123,13323],"mathbb","”。",[18,13326,13327,13473,14114,14516],{},[21,13328,13329,55,13332,13472],{},[24,13330,13331],{},"设定",[57,13333,13335,13365],{"className":13334},[60],[57,13336,13338],{"className":13337},[64],[66,13339,13340],{"xmlns":68},[70,13341,13342,13362],{},[73,13343,13344,13350,13352,13354,13356],{},[80,13345,13346,13348],{},[83,13347,14],{},[278,13349,280],{},[76,13351,283],{"separator":98},[76,13353,506],{},[76,13355,283],{"separator":98},[80,13357,13358,13360],{},[83,13359,14],{},[83,13361,2206],{},[90,13363,13364],{"encoding":92},"p_1, \\dots, p_n",[57,13366,13368],{"className":13367,"ariaHidden":98},[97],[57,13369,13371,13374,13414,13417,13420,13423,13426,13429,13432],{"className":13370},[102],[57,13372],{"className":13373,"style":3148},[106],[57,13375,13377,13380],{"className":13376},[123],[57,13378,14],{"className":13379},[123,127],[57,13381,13383],{"className":13382},[132],[57,13384,13386,13406],{"className":13385},[136,137],[57,13387,13389,13403],{"className":13388},[141],[57,13390,13392],{"className":13391,"style":339},[145],[57,13393,13394,13397],{"style":550},[57,13395],{"className":13396,"style":154},[153],[57,13398,13400],{"className":13399},[158,159,160,161],[57,13401,280],{"className":13402},[123,161],[57,13404,169],{"className":13405},[168],[57,13407,13409],{"className":13408},[141],[57,13410,13412],{"className":13411,"style":176},[145],[57,13413],{},[57,13415,283],{"className":13416},[365],[57,13418],{"className":13419,"style":119},[118],[57,13421,506],{"className":13422},[672],[57,13424],{"className":13425,"style":119},[118],[57,13427,283],{"className":13428},[365],[57,13430],{"className":13431,"style":119},[118],[57,13433,13435,13438],{"className":13434},[123],[57,13436,14],{"className":13437},[123,127],[57,13439,13441],{"className":13440},[132],[57,13442,13444,13464],{"className":13443},[136,137],[57,13445,13447,13461],{"className":13446},[141],[57,13448,13450],{"className":13449,"style":2295},[145],[57,13451,13452,13455],{"style":550},[57,13453],{"className":13454,"style":154},[153],[57,13456,13458],{"className":13457},[158,159,160,161],[57,13459,2206],{"className":13460},[123,127,161],[57,13462,169],{"className":13463},[168],[57,13465,13467],{"className":13466},[141],[57,13468,13470],{"className":13469,"style":176},[145],[57,13471],{}," 是两两互素的整数（比如 3, 5, 7，不一定是质数，只要最大公约数是1即可）。",[21,13474,13475,5141,13478],{},[24,13476,13477],{},"对应关系",[184,13479,13480,13571,13819],{},[21,13481,13482,13483,13511,13512,13542,13543,3906],{},"环 ",[57,13484,13486,13499],{"className":13485},[60],[57,13487,13489],{"className":13488},[64],[66,13490,13491],{"xmlns":68},[70,13492,13493,13497],{},[73,13494,13495],{},[83,13496,85],{},[90,13498,85],{"encoding":92},[57,13500,13502],{"className":13501,"ariaHidden":98},[97],[57,13503,13505,13508],{"className":13504},[102],[57,13506],{"className":13507,"style":2507},[106],[57,13509,85],{"className":13510,"style":128},[123,127]," ",[57,13513,13515,13529],{"className":13514},[60],[57,13516,13518],{"className":13517},[64],[66,13519,13520],{"xmlns":68},[70,13521,13522,13526],{},[73,13523,13524],{},[76,13525,3204],{},[90,13527,13528],{"encoding":92},"\\rightarrow",[57,13530,13532],{"className":13531,"ariaHidden":98},[97],[57,13533,13535,13539],{"className":13534},[102],[57,13536],{"className":13537,"style":13538},[106],"height:0.3669em;",[57,13540,3204],{"className":13541},[1285]," 整数集 ",[57,13544,13546,13559],{"className":13545},[60],[57,13547,13549],{"className":13548},[64],[66,13550,13551],{"xmlns":68},[70,13552,13553,13557],{},[73,13554,13555],{},[83,13556,13306],{"mathvariant":13305},[90,13558,13309],{"encoding":92},[57,13560,13562],{"className":13561,"ariaHidden":98},[97],[57,13563,13565,13568],{"className":13564},[102],[57,13566],{"className":13567,"style":13319},[106],[57,13569,13306],{"className":13570},[123,13323],[21,13572,13573,13574,13511,13643,13511,13671,13747,13748,13818],{},"理想 ",[57,13575,13577,13594],{"className":13576},[60],[57,13578,13580],{"className":13579},[64],[66,13581,13582],{"xmlns":68},[70,13583,13584,13592],{},[73,13585,13586],{},[80,13587,13588,13590],{},[83,13589,2354],{},[83,13591,2545],{},[90,13593,12089],{"encoding":92},[57,13595,13597],{"className":13596,"ariaHidden":98},[97],[57,13598,13600,13603],{"className":13599},[102],[57,13601],{"className":13602,"style":219},[106],[57,13604,13606,13609],{"className":13605},[123],[57,13607,2354],{"className":13608,"style":2387},[123,127],[57,13610,13612],{"className":13611},[132],[57,13613,13615,13635],{"className":13614},[136,137],[57,13616,13618,13632],{"className":13617},[141],[57,13619,13621],{"className":13620,"style":2695},[145],[57,13622,13623,13626],{"style":2402},[57,13624],{"className":13625,"style":154},[153],[57,13627,13629],{"className":13628},[158,159,160,161],[57,13630,2545],{"className":13631,"style":2643},[123,127,161],[57,13633,169],{"className":13634},[168],[57,13636,13638],{"className":13637},[141],[57,13639,13641],{"className":13640,"style":176},[145],[57,13642],{},[57,13644,13646,13659],{"className":13645},[60],[57,13647,13649],{"className":13648},[64],[66,13650,13651],{"xmlns":68},[70,13652,13653,13657],{},[73,13654,13655],{},[76,13656,3204],{},[90,13658,13528],{"encoding":92},[57,13660,13662],{"className":13661,"ariaHidden":98},[97],[57,13663,13665,13668],{"className":13664},[102],[57,13666],{"className":13667,"style":13538},[106],[57,13669,3204],{"className":13670},[1285],[57,13672,13674,13694],{"className":13673},[60],[57,13675,13677],{"className":13676},[64],[66,13678,13679],{"xmlns":68},[70,13680,13681,13691],{},[73,13682,13683,13689],{},[80,13684,13685,13687],{},[83,13686,14],{},[83,13688,2545],{},[83,13690,13306],{"mathvariant":13305},[90,13692,13693],{"encoding":92},"p_k \\mathbb{Z}",[57,13695,13697],{"className":13696,"ariaHidden":98},[97],[57,13698,13700,13704,13744],{"className":13699},[102],[57,13701],{"className":13702,"style":13703},[106],"height:0.8833em;vertical-align:-0.1944em;",[57,13705,13707,13710],{"className":13706},[123],[57,13708,14],{"className":13709},[123,127],[57,13711,13713],{"className":13712},[132],[57,13714,13716,13736],{"className":13715},[136,137],[57,13717,13719,13733],{"className":13718},[141],[57,13720,13722],{"className":13721,"style":2695},[145],[57,13723,13724,13727],{"style":550},[57,13725],{"className":13726,"style":154},[153],[57,13728,13730],{"className":13729},[158,159,160,161],[57,13731,2545],{"className":13732,"style":2643},[123,127,161],[57,13734,169],{"className":13735},[168],[57,13737,13739],{"className":13738},[141],[57,13740,13742],{"className":13741,"style":176},[145],[57,13743],{},[57,13745,13306],{"className":13746},[123,13323],"（",[57,13749,13751,13769],{"className":13750},[60],[57,13752,13754],{"className":13753},[64],[66,13755,13756],{"xmlns":68},[70,13757,13758,13766],{},[73,13759,13760],{},[80,13761,13762,13764],{},[83,13763,14],{},[83,13765,2545],{},[90,13767,13768],{"encoding":92},"p_k",[57,13770,13772],{"className":13771,"ariaHidden":98},[97],[57,13773,13775,13778],{"className":13774},[102],[57,13776],{"className":13777,"style":3148},[106],[57,13779,13781,13784],{"className":13780},[123],[57,13782,14],{"className":13783},[123,127],[57,13785,13787],{"className":13786},[132],[57,13788,13790,13810],{"className":13789},[136,137],[57,13791,13793,13807],{"className":13792},[141],[57,13794,13796],{"className":13795,"style":2695},[145],[57,13797,13798,13801],{"style":550},[57,13799],{"className":13800,"style":154},[153],[57,13802,13804],{"className":13803},[158,159,160,161],[57,13805,2545],{"className":13806,"style":2643},[123,127,161],[57,13808,169],{"className":13809},[168],[57,13811,13813],{"className":13812},[141],[57,13814,13816],{"className":13815,"style":176},[145],[57,13817],{}," 的倍数）。",[21,13820,13821,13822,13511,13897,13925,13926,14020,14021,3906],{},"交集 ",[57,13823,13825,13845],{"className":13824},[60],[57,13826,13828],{"className":13827},[64],[66,13829,13830],{"xmlns":68},[70,13831,13832,13842],{},[73,13833,13834,13836],{},[76,13835,4472],{},[80,13837,13838,13840],{},[83,13839,2354],{},[83,13841,2545],{},[90,13843,13844],{"encoding":92},"\\cap I_k",[57,13846,13848],{"className":13847,"ariaHidden":98},[97],[57,13849,13851,13854,13857],{"className":13850},[102],[57,13852],{"className":13853,"style":219},[106],[57,13855,4472],{"className":13856},[123],[57,13858,13860,13863],{"className":13859},[123],[57,13861,2354],{"className":13862,"style":2387},[123,127],[57,13864,13866],{"className":13865},[132],[57,13867,13869,13889],{"className":13868},[136,137],[57,13870,13872,13886],{"className":13871},[141],[57,13873,13875],{"className":13874,"style":2695},[145],[57,13876,13877,13880],{"style":2402},[57,13878],{"className":13879,"style":154},[153],[57,13881,13883],{"className":13882},[158,159,160,161],[57,13884,2545],{"className":13885,"style":2643},[123,127,161],[57,13887,169],{"className":13888},[168],[57,13890,13892],{"className":13891},[141],[57,13893,13895],{"className":13894,"style":176},[145],[57,13896],{},[57,13898,13900,13913],{"className":13899},[60],[57,13901,13903],{"className":13902},[64],[66,13904,13905],{"xmlns":68},[70,13906,13907,13911],{},[73,13908,13909],{},[76,13910,3204],{},[90,13912,13528],{"encoding":92},[57,13914,13916],{"className":13915,"ariaHidden":98},[97],[57,13917,13919,13922],{"className":13918},[102],[57,13920],{"className":13921,"style":13538},[106],[57,13923,3204],{"className":13924},[1285]," 最小公倍数 ",[57,13927,13929,13956],{"className":13928},[60],[57,13930,13932],{"className":13931},[64],[66,13933,13934],{"xmlns":68},[70,13935,13936,13953],{},[73,13937,13938,13941,13943,13949,13951],{},[508,13939,13940],{},"lcm",[76,13942,478],{"stretchy":477},[80,13944,13945,13947],{},[83,13946,14],{},[83,13948,88],{},[76,13950,513],{"stretchy":477},[83,13952,13306],{"mathvariant":13305},[90,13954,13955],{"encoding":92},"\\text{lcm}(p_i)\\mathbb{Z}",[57,13957,13959],{"className":13958,"ariaHidden":98},[97],[57,13960,13962,13965,13971,13974,14014,14017],{"className":13961},[102],[57,13963],{"className":13964,"style":107},[106],[57,13966,13968],{"className":13967},[123,4851],[57,13969,13940],{"className":13970},[123],[57,13972,478],{"className":13973},[529],[57,13975,13977,13980],{"className":13976},[123],[57,13978,14],{"className":13979},[123,127],[57,13981,13983],{"className":13982},[132],[57,13984,13986,14006],{"className":13985},[136,137],[57,13987,13989,14003],{"className":13988},[141],[57,13990,13992],{"className":13991,"style":146},[145],[57,13993,13994,13997],{"style":550},[57,13995],{"className":13996,"style":154},[153],[57,13998,14000],{"className":13999},[158,159,160,161],[57,14001,88],{"className":14002},[123,127,161],[57,14004,169],{"className":14005},[168],[57,14007,14009],{"className":14008},[141],[57,14010,14012],{"className":14011,"style":176},[145],[57,14013],{},[57,14015,513],{"className":14016},[679],[57,14018,13306],{"className":14019},[123,13323],"。因为互素，这就是乘积 ",[57,14022,14024,14050],{"className":14023},[60],[57,14025,14027],{"className":14026},[64],[66,14028,14029],{"xmlns":68},[70,14030,14031,14047],{},[73,14032,14033,14035,14037,14043,14045],{},[76,14034,478],{"stretchy":477},[76,14036,78],{},[80,14038,14039,14041],{},[83,14040,14],{},[83,14042,88],{},[76,14044,513],{"stretchy":477},[83,14046,13306],{"mathvariant":13305},[90,14048,14049],{"encoding":92},"(\\prod p_i)\\mathbb{Z}",[57,14051,14053],{"className":14052,"ariaHidden":98},[97],[57,14054,14056,14059,14062,14065,14068,14108,14111],{"className":14055},[102],[57,14057],{"className":14058,"style":107},[106],[57,14060,478],{"className":14061},[529],[57,14063,78],{"className":14064,"style":114},[111,112,113],[57,14066],{"className":14067,"style":119},[118],[57,14069,14071,14074],{"className":14070},[123],[57,14072,14],{"className":14073},[123,127],[57,14075,14077],{"className":14076},[132],[57,14078,14080,14100],{"className":14079},[136,137],[57,14081,14083,14097],{"className":14082},[141],[57,14084,14086],{"className":14085,"style":146},[145],[57,14087,14088,14091],{"style":550},[57,14089],{"className":14090,"style":154},[153],[57,14092,14094],{"className":14093},[158,159,160,161],[57,14095,88],{"className":14096},[123,127,161],[57,14098,169],{"className":14099},[168],[57,14101,14103],{"className":14102},[141],[57,14104,14106],{"className":14105,"style":176},[145],[57,14107],{},[57,14109,513],{"className":14110},[679],[57,14112,13306],{"className":14113},[123,13323],[21,14115,14116,5141,14119,14305,14306,14334,14335,14364,14365,14515],{},[24,14117,14118],{},"方程组",[57,14120,14122,14168],{"className":14121},[60],[57,14123,14125],{"className":14124},[64],[66,14126,14127],{"xmlns":68},[70,14128,14129,14165],{},[73,14130,14131,14133,14135,14141,14143,14145,14147,14155,14157,14163],{},[83,14132,3000],{},[76,14134,3926],{},[80,14136,14137,14139],{},[83,14138,483],{},[83,14140,88],{},[118,14142],{},[118,14144],{"width":3934},[76,14146,478],{"stretchy":477},[73,14148,14149,14151,14153],{},[83,14150,3941],{"mathvariant":302},[83,14152,3944],{"mathvariant":302},[83,14154,3947],{"mathvariant":302},[118,14156],{"width":3950},[80,14158,14159,14161],{},[83,14160,14],{},[83,14162,88],{},[76,14164,513],{"stretchy":477},[90,14166,14167],{"encoding":92},"x \\equiv a_i \\pmod{p_i}",[57,14169,14171,14189,14241],{"className":14170,"ariaHidden":98},[97],[57,14172,14174,14177,14180,14183,14186],{"className":14173},[102],[57,14175],{"className":14176,"style":3971},[106],[57,14178,3000],{"className":14179},[123,127],[57,14181],{"className":14182,"style":1281},[118],[57,14184,3926],{"className":14185},[1285],[57,14187],{"className":14188,"style":1281},[118],[57,14190,14192,14195,14235,14238],{"className":14191},[102],[57,14193],{"className":14194,"style":711},[106],[57,14196,14198,14201],{"className":14197},[123],[57,14199,483],{"className":14200},[123,127],[57,14202,14204],{"className":14203},[132],[57,14205,14207,14227],{"className":14206},[136,137],[57,14208,14210,14224],{"className":14209},[141],[57,14211,14213],{"className":14212,"style":146},[145],[57,14214,14215,14218],{"style":550},[57,14216],{"className":14217,"style":154},[153],[57,14219,14221],{"className":14220},[158,159,160,161],[57,14222,88],{"className":14223},[123,127,161],[57,14225,169],{"className":14226},[168],[57,14228,14230],{"className":14229},[141],[57,14231,14233],{"className":14232,"style":176},[145],[57,14234],{},[57,14236],{"className":14237},[118,3996],[57,14239],{"className":14240,"style":4000},[118],[57,14242,14244,14247,14250,14259,14262,14302],{"className":14243},[102],[57,14245],{"className":14246,"style":107},[106],[57,14248,478],{"className":14249},[529],[57,14251,14253],{"className":14252},[123],[57,14254,14256],{"className":14255},[123],[57,14257,4020],{"className":14258},[123,4019],[57,14260],{"className":14261,"style":4024},[118],[57,14263,14265,14268],{"className":14264},[123],[57,14266,14],{"className":14267},[123,127],[57,14269,14271],{"className":14270},[132],[57,14272,14274,14294],{"className":14273},[136,137],[57,14275,14277,14291],{"className":14276},[141],[57,14278,14280],{"className":14279,"style":146},[145],[57,14281,14282,14285],{"style":550},[57,14283],{"className":14284,"style":154},[153],[57,14286,14288],{"className":14287},[158,159,160,161],[57,14289,88],{"className":14290},[123,127,161],[57,14292,169],{"className":14293},[168],[57,14295,14297],{"className":14296},[141],[57,14298,14300],{"className":14299,"style":176},[145],[57,14301],{},[57,14303,513],{"className":14304},[679],"\n这其实就是在问：有没有一个整数 ",[57,14307,14309,14322],{"className":14308},[60],[57,14310,14312],{"className":14311},[64],[66,14313,14314],{"xmlns":68},[70,14315,14316,14320],{},[73,14317,14318],{},[83,14319,3000],{},[90,14321,3000],{"encoding":92},[57,14323,14325],{"className":14324,"ariaHidden":98},[97],[57,14326,14328,14331],{"className":14327},[102],[57,14329],{"className":14330,"style":3686},[106],[57,14332,3000],{"className":14333},[123,127],"，经过映射 ",[57,14336,14338,14352],{"className":14337},[60],[57,14339,14341],{"className":14340},[64],[66,14342,14343],{"xmlns":68},[70,14344,14345,14349],{},[73,14346,14347],{},[83,14348,3196],{},[90,14350,14351],{"encoding":92},"\\phi",[57,14353,14355],{"className":14354,"ariaHidden":98},[97],[57,14356,14358,14361],{"className":14357},[102],[57,14359],{"className":14360,"style":1353},[106],[57,14362,3196],{"className":14363},[123,127]," 后，变成了 ",[57,14366,14368,14402],{"className":14367},[60],[57,14369,14371],{"className":14370},[64],[66,14372,14373],{"xmlns":68},[70,14374,14375,14399],{},[73,14376,14377,14379,14385,14387,14389,14391,14397],{},[76,14378,478],{"stretchy":477},[80,14380,14381,14383],{},[83,14382,483],{},[278,14384,280],{},[76,14386,283],{"separator":98},[76,14388,506],{},[76,14390,283],{"separator":98},[80,14392,14393,14395],{},[83,14394,483],{},[83,14396,2206],{},[76,14398,513],{"stretchy":477},[90,14400,14401],{"encoding":92},"(a_1, \\dots, a_n)",[57,14403,14405],{"className":14404,"ariaHidden":98},[97],[57,14406,14408,14411,14414,14454,14457,14460,14463,14466,14469,14472,14512],{"className":14407},[102],[57,14409],{"className":14410,"style":107},[106],[57,14412,478],{"className":14413},[529],[57,14415,14417,14420],{"className":14416},[123],[57,14418,483],{"className":14419},[123,127],[57,14421,14423],{"className":14422},[132],[57,14424,14426,14446],{"className":14425},[136,137],[57,14427,14429,14443],{"className":14428},[141],[57,14430,14432],{"className":14431,"style":339},[145],[57,14433,14434,14437],{"style":550},[57,14435],{"className":14436,"style":154},[153],[57,14438,14440],{"className":14439},[158,159,160,161],[57,14441,280],{"className":14442},[123,161],[57,14444,169],{"className":14445},[168],[57,14447,14449],{"className":14448},[141],[57,14450,14452],{"className":14451,"style":176},[145],[57,14453],{},[57,14455,283],{"className":14456},[365],[57,14458],{"className":14459,"style":119},[118],[57,14461,506],{"className":14462},[672],[57,14464],{"className":14465,"style":119},[118],[57,14467,283],{"className":14468},[365],[57,14470],{"className":14471,"style":119},[118],[57,14473,14475,14478],{"className":14474},[123],[57,14476,483],{"className":14477},[123,127],[57,14479,14481],{"className":14480},[132],[57,14482,14484,14504],{"className":14483},[136,137],[57,14485,14487,14501],{"className":14486},[141],[57,14488,14490],{"className":14489,"style":2295},[145],[57,14491,14492,14495],{"style":550},[57,14493],{"className":14494,"style":154},[153],[57,14496,14498],{"className":14497},[158,159,160,161],[57,14499,2206],{"className":14500},[123,127,161],[57,14502,169],{"className":14503},[168],[57,14505,14507],{"className":14506},[141],[57,14508,14510],{"className":14509,"style":176},[145],[57,14511],{},[57,14513,513],{"className":14514},[679],"？",[21,14517,14518,14521,14522,14525,14526],{},[24,14519,14520],{},"结论","：\n根据上面的定理，映射是",[24,14523,14524],{},"满射","（Surjective）。\n",[184,14527,14528,14636,14809],{},[21,14529,14530,14532,14533,14536,14537,14607,14608,3906],{},[24,14531,14524],{},"意味着：",[24,14534,14535],{},"总有解","。无论 ",[57,14538,14540,14558],{"className":14539},[60],[57,14541,14543],{"className":14542},[64],[66,14544,14545],{"xmlns":68},[70,14546,14547,14555],{},[73,14548,14549],{},[80,14550,14551,14553],{},[83,14552,483],{},[83,14554,88],{},[90,14556,14557],{"encoding":92},"a_i",[57,14559,14561],{"className":14560,"ariaHidden":98},[97],[57,14562,14564,14567],{"className":14563},[102],[57,14565],{"className":14566,"style":711},[106],[57,14568,14570,14573],{"className":14569},[123],[57,14571,483],{"className":14572},[123,127],[57,14574,14576],{"className":14575},[132],[57,14577,14579,14599],{"className":14578},[136,137],[57,14580,14582,14596],{"className":14581},[141],[57,14583,14585],{"className":14584,"style":146},[145],[57,14586,14587,14590],{"style":550},[57,14588],{"className":14589,"style":154},[153],[57,14591,14593],{"className":14592},[158,159,160,161],[57,14594,88],{"className":14595},[123,127,161],[57,14597,169],{"className":14598},[168],[57,14600,14602],{"className":14601},[141],[57,14603,14605],{"className":14604,"style":176},[145],[57,14606],{}," 是多少，都能找到 ",[57,14609,14611,14624],{"className":14610},[60],[57,14612,14614],{"className":14613},[64],[66,14615,14616],{"xmlns":68},[70,14617,14618,14622],{},[73,14619,14620],{},[83,14621,3000],{},[90,14623,3000],{"encoding":92},[57,14625,14627],{"className":14626,"ariaHidden":98},[97],[57,14628,14630,14633],{"className":14629},[102],[57,14631],{"className":14632,"style":3686},[106],[57,14634,3000],{"className":14635},[123,127],[21,14637,14638,14641,14642,14725,14726,14804,14805,14808],{},[24,14639,14640],{},"核","是 ",[57,14643,14645,14667],{"className":14644},[60],[57,14646,14648],{"className":14647},[64],[66,14649,14650],{"xmlns":68},[70,14651,14652,14664],{},[73,14653,14654,14656,14662],{},[76,14655,78],{},[80,14657,14658,14660],{},[83,14659,14],{},[83,14661,88],{},[83,14663,13306],{"mathvariant":13305},[90,14665,14666],{"encoding":92},"\\prod p_i \\mathbb{Z}",[57,14668,14670],{"className":14669,"ariaHidden":98},[97],[57,14671,14673,14676,14679,14682,14722],{"className":14672},[102],[57,14674],{"className":14675,"style":107},[106],[57,14677,78],{"className":14678,"style":114},[111,112,113],[57,14680],{"className":14681,"style":119},[118],[57,14683,14685,14688],{"className":14684},[123],[57,14686,14],{"className":14687},[123,127],[57,14689,14691],{"className":14690},[132],[57,14692,14694,14714],{"className":14693},[136,137],[57,14695,14697,14711],{"className":14696},[141],[57,14698,14700],{"className":14699,"style":146},[145],[57,14701,14702,14705],{"style":550},[57,14703],{"className":14704,"style":154},[153],[57,14706,14708],{"className":14707},[158,159,160,161],[57,14709,88],{"className":14710},[123,127,161],[57,14712,169],{"className":14713},[168],[57,14715,14717],{"className":14716},[141],[57,14718,14720],{"className":14719,"style":176},[145],[57,14721],{},[57,14723,13306],{"className":14724},[123,13323],"：这意味着解在模 ",[57,14727,14729,14749],{"className":14728},[60],[57,14730,14732],{"className":14731},[64],[66,14733,14734],{"xmlns":68},[70,14735,14736,14746],{},[73,14737,14738,14740],{},[76,14739,78],{},[80,14741,14742,14744],{},[83,14743,14],{},[83,14745,88],{},[90,14747,14748],{"encoding":92},"\\prod p_i",[57,14750,14752],{"className":14751,"ariaHidden":98},[97],[57,14753,14755,14758,14761,14764],{"className":14754},[102],[57,14756],{"className":14757,"style":107},[106],[57,14759,78],{"className":14760,"style":114},[111,112,113],[57,14762],{"className":14763,"style":119},[118],[57,14765,14767,14770],{"className":14766},[123],[57,14768,14],{"className":14769},[123,127],[57,14771,14773],{"className":14772},[132],[57,14774,14776,14796],{"className":14775},[136,137],[57,14777,14779,14793],{"className":14778},[141],[57,14780,14782],{"className":14781,"style":146},[145],[57,14783,14784,14787],{"style":550},[57,14785],{"className":14786,"style":154},[153],[57,14788,14790],{"className":14789},[158,159,160,161],[57,14791,88],{"className":14792},[123,127,161],[57,14794,169],{"className":14795},[168],[57,14797,14799],{"className":14798},[141],[57,14800,14802],{"className":14801,"style":176},[145],[57,14803],{}," 的意义下是",[24,14806,14807],{},"唯一","的。",[21,14810,14811,14812,14933,14934,2879,14962,14990],{},"推论里写的 ",[57,14813,14815,14848],{"className":14814},[60],[57,14816,14818],{"className":14817},[64],[66,14819,14820],{"xmlns":68},[70,14821,14822,14845],{},[73,14823,14824,14826,14832,14835,14837,14839,14841,14843],{},[76,14825,78],{},[80,14827,14828,14830],{},[83,14829,14],{},[83,14831,88],{},[83,14833,14834],{"mathvariant":302},"∣",[76,14836,478],{"stretchy":477},[83,14838,3000],{},[76,14840,7706],{},[83,14842,3054],{},[76,14844,513],{"stretchy":477},[90,14846,14847],{"encoding":92},"\\prod p_i | (x-y)",[57,14849,14851,14921],{"className":14850,"ariaHidden":98},[97],[57,14852,14854,14857,14860,14863,14903,14906,14909,14912,14915,14918],{"className":14853},[102],[57,14855],{"className":14856,"style":107},[106],[57,14858,78],{"className":14859,"style":114},[111,112,113],[57,14861],{"className":14862,"style":119},[118],[57,14864,14866,14869],{"className":14865},[123],[57,14867,14],{"className":14868},[123,127],[57,14870,14872],{"className":14871},[132],[57,14873,14875,14895],{"className":14874},[136,137],[57,14876,14878,14892],{"className":14877},[141],[57,14879,14881],{"className":14880,"style":146},[145],[57,14882,14883,14886],{"style":550},[57,14884],{"className":14885,"style":154},[153],[57,14887,14889],{"className":14888},[158,159,160,161],[57,14890,88],{"className":14891},[123,127,161],[57,14893,169],{"className":14894},[168],[57,14896,14898],{"className":14897},[141],[57,14899,14901],{"className":14900,"style":176},[145],[57,14902],{},[57,14904,14834],{"className":14905},[123],[57,14907,478],{"className":14908},[529],[57,14910,3000],{"className":14911},[123,127],[57,14913],{"className":14914,"style":1172},[118],[57,14916,7706],{"className":14917},[1176],[57,14919],{"className":14920,"style":1172},[118],[57,14922,14924,14927,14930],{"className":14923},[102],[57,14925],{"className":14926,"style":107},[106],[57,14928,3054],{"className":14929,"style":3075},[123,127],[57,14931,513],{"className":14932},[679]," 就是指：如果有两个解 ",[57,14935,14937,14950],{"className":14936},[60],[57,14938,14940],{"className":14939},[64],[66,14941,14942],{"xmlns":68},[70,14943,14944,14948],{},[73,14945,14946],{},[83,14947,3000],{},[90,14949,3000],{"encoding":92},[57,14951,14953],{"className":14952,"ariaHidden":98},[97],[57,14954,14956,14959],{"className":14955},[102],[57,14957],{"className":14958,"style":3686},[106],[57,14960,3000],{"className":14961},[123,127],[57,14963,14965,14978],{"className":14964},[60],[57,14966,14968],{"className":14967},[64],[66,14969,14970],{"xmlns":68},[70,14971,14972,14976],{},[73,14973,14974],{},[83,14975,3054],{},[90,14977,3054],{"encoding":92},[57,14979,14981],{"className":14980,"ariaHidden":98},[97],[57,14982,14984,14987],{"className":14983},[102],[57,14985],{"className":14986,"style":3148},[106],[57,14988,3054],{"className":14989,"style":3075},[123,127],"，它们的差必须能被总乘积整除（也就是它们其实是同一个解）。",[35,14992],{},[38,14994,14995],{"id":14995},"全局总结",[18,14997,14998,15004,15010],{},[21,14999,15000,15003],{},[24,15001,15002],{},"直积\u002F直和","：就是把环拼起来的容器。有限个时它俩一样。",[21,15005,15006,15009],{},[24,15007,15008],{},"中国剩余定理 (CRT)","：告诉我们，如果几个理想互不干扰（互素），那么“整体模交集”就完全等同于“分开模各自分量”。",[21,15011,15012,15015,15016],{},[24,15013,15014],{},"证明的核心技巧","：如何证明你能控制每一个分量？\n",[184,15017,15018,15164,15306,15448],{},[21,15019,15020,15021,13511,15024,3906],{},"你需要找到一组",[24,15022,15023],{},"开关",[57,15025,15027,15057],{"className":15026},[60],[57,15028,15030],{"className":15029},[64],[66,15031,15032],{"xmlns":68},[70,15033,15034,15054],{},[73,15035,15036,15042,15044,15046,15048],{},[80,15037,15038,15040],{},[83,15039,3941],{},[278,15041,280],{},[76,15043,283],{"separator":98},[76,15045,506],{},[76,15047,283],{"separator":98},[80,15049,15050,15052],{},[83,15051,3941],{},[83,15053,2206],{},[90,15055,15056],{"encoding":92},"m_1, \\dots, m_n",[57,15058,15060],{"className":15059,"ariaHidden":98},[97],[57,15061,15063,15066,15106,15109,15112,15115,15118,15121,15124],{"className":15062},[102],[57,15064],{"className":15065,"style":3148},[106],[57,15067,15069,15072],{"className":15068},[123],[57,15070,3941],{"className":15071},[123,127],[57,15073,15075],{"className":15074},[132],[57,15076,15078,15098],{"className":15077},[136,137],[57,15079,15081,15095],{"className":15080},[141],[57,15082,15084],{"className":15083,"style":339},[145],[57,15085,15086,15089],{"style":550},[57,15087],{"className":15088,"style":154},[153],[57,15090,15092],{"className":15091},[158,159,160,161],[57,15093,280],{"className":15094},[123,161],[57,15096,169],{"className":15097},[168],[57,15099,15101],{"className":15100},[141],[57,15102,15104],{"className":15103,"style":176},[145],[57,15105],{},[57,15107,283],{"className":15108},[365],[57,15110],{"className":15111,"style":119},[118],[57,15113,506],{"className":15114},[672],[57,15116],{"className":15117,"style":119},[118],[57,15119,283],{"className":15120},[365],[57,15122],{"className":15123,"style":119},[118],[57,15125,15127,15130],{"className":15126},[123],[57,15128,3941],{"className":15129},[123,127],[57,15131,15133],{"className":15132},[132],[57,15134,15136,15156],{"className":15135},[136,137],[57,15137,15139,15153],{"className":15138},[141],[57,15140,15142],{"className":15141,"style":2295},[145],[57,15143,15144,15147],{"style":550},[57,15145],{"className":15146,"style":154},[153],[57,15148,15150],{"className":15149},[158,159,160,161],[57,15151,2206],{"className":15152},[123,127,161],[57,15154,169],{"className":15155},[168],[57,15157,15159],{"className":15158},[141],[57,15160,15162],{"className":15161,"style":176},[145],[57,15163],{},[21,15165,15166,15235,15236,15305],{},[57,15167,15169,15186],{"className":15168},[60],[57,15170,15172],{"className":15171},[64],[66,15173,15174],{"xmlns":68},[70,15175,15176,15184],{},[73,15177,15178],{},[80,15179,15180,15182],{},[83,15181,3941],{},[278,15183,280],{},[90,15185,9226],{"encoding":92},[57,15187,15189],{"className":15188,"ariaHidden":98},[97],[57,15190,15192,15195],{"className":15191},[102],[57,15193],{"className":15194,"style":711},[106],[57,15196,15198,15201],{"className":15197},[123],[57,15199,3941],{"className":15200},[123,127],[57,15202,15204],{"className":15203},[132],[57,15205,15207,15227],{"className":15206},[136,137],[57,15208,15210,15224],{"className":15209},[141],[57,15211,15213],{"className":15212,"style":339},[145],[57,15214,15215,15218],{"style":550},[57,15216],{"className":15217,"style":154},[153],[57,15219,15221],{"className":15220},[158,159,160,161],[57,15222,280],{"className":15223},[123,161],[57,15225,169],{"className":15226},[168],[57,15228,15230],{"className":15229},[141],[57,15231,15233],{"className":15232,"style":176},[145],[57,15234],{}," 负责 ",[57,15237,15239,15256],{"className":15238},[60],[57,15240,15242],{"className":15241},[64],[66,15243,15244],{"xmlns":68},[70,15245,15246,15254],{},[73,15247,15248],{},[80,15249,15250,15252],{},[83,15251,2354],{},[278,15253,280],{},[90,15255,5005],{"encoding":92},[57,15257,15259],{"className":15258,"ariaHidden":98},[97],[57,15260,15262,15265],{"className":15261},[102],[57,15263],{"className":15264,"style":219},[106],[57,15266,15268,15271],{"className":15267},[123],[57,15269,2354],{"className":15270,"style":2387},[123,127],[57,15272,15274],{"className":15273},[132],[57,15275,15277,15297],{"className":15276},[136,137],[57,15278,15280,15294],{"className":15279},[141],[57,15281,15283],{"className":15282,"style":339},[145],[57,15284,15285,15288],{"style":2402},[57,15286],{"className":15287,"style":154},[153],[57,15289,15291],{"className":15290},[158,159,160,161],[57,15292,280],{"className":15293},[123,161],[57,15295,169],{"className":15296},[168],[57,15298,15300],{"className":15299},[141],[57,15301,15303],{"className":15302,"style":176},[145],[57,15304],{},"（在这里是1，在别处是0）。",[21,15307,15308,15235,15378,15305],{},[57,15309,15311,15329],{"className":15310},[60],[57,15312,15314],{"className":15313},[64],[66,15315,15316],{"xmlns":68},[70,15317,15318,15326],{},[73,15319,15320],{},[80,15321,15322,15324],{},[83,15323,3941],{},[278,15325,290],{},[90,15327,15328],{"encoding":92},"m_2",[57,15330,15332],{"className":15331,"ariaHidden":98},[97],[57,15333,15335,15338],{"className":15334},[102],[57,15336],{"className":15337,"style":711},[106],[57,15339,15341,15344],{"className":15340},[123],[57,15342,3941],{"className":15343},[123,127],[57,15345,15347],{"className":15346},[132],[57,15348,15350,15370],{"className":15349},[136,137],[57,15351,15353,15367],{"className":15352},[141],[57,15354,15356],{"className":15355,"style":339},[145],[57,15357,15358,15361],{"style":550},[57,15359],{"className":15360,"style":154},[153],[57,15362,15364],{"className":15363},[158,159,160,161],[57,15365,290],{"className":15366},[123,161],[57,15368,169],{"className":15369},[168],[57,15371,15373],{"className":15372},[141],[57,15374,15376],{"className":15375,"style":176},[145],[57,15377],{},[57,15379,15381,15399],{"className":15380},[60],[57,15382,15384],{"className":15383},[64],[66,15385,15386],{"xmlns":68},[70,15387,15388,15396],{},[73,15389,15390],{},[80,15391,15392,15394],{},[83,15393,2354],{},[278,15395,290],{},[90,15397,15398],{"encoding":92},"I_2",[57,15400,15402],{"className":15401,"ariaHidden":98},[97],[57,15403,15405,15408],{"className":15404},[102],[57,15406],{"className":15407,"style":219},[106],[57,15409,15411,15414],{"className":15410},[123],[57,15412,2354],{"className":15413,"style":2387},[123,127],[57,15415,15417],{"className":15416},[132],[57,15418,15420,15440],{"className":15419},[136,137],[57,15421,15423,15437],{"className":15422},[141],[57,15424,15426],{"className":15425,"style":339},[145],[57,15427,15428,15431],{"style":2402},[57,15429],{"className":15430,"style":154},[153],[57,15432,15434],{"className":15433},[158,159,160,161],[57,15435,290],{"className":15436},[123,161],[57,15438,169],{"className":15439},[168],[57,15441,15443],{"className":15442},[141],[57,15444,15446],{"className":15445,"style":176},[145],[57,15447],{},[21,15449,15450,15451,15597],{},"然后用 ",[57,15452,15454,15484],{"className":15453},[60],[57,15455,15457],{"className":15456},[64],[66,15458,15459],{"xmlns":68},[70,15460,15461,15481],{},[73,15462,15463,15465,15467,15469,15475],{},[83,15464,3000],{},[76,15466,1030],{},[76,15468,8729],{},[80,15470,15471,15473],{},[83,15472,3941],{},[83,15474,88],{},[80,15476,15477,15479],{},[83,15478,3000],{},[83,15480,88],{},[90,15482,15483],{"encoding":92},"x = \\sum m_i x_i",[57,15485,15487,15505],{"className":15486,"ariaHidden":98},[97],[57,15488,15490,15493,15496,15499,15502],{"className":15489},[102],[57,15491],{"className":15492,"style":3686},[106],[57,15494,3000],{"className":15495},[123,127],[57,15497],{"className":15498,"style":1281},[118],[57,15500,1030],{"className":15501},[1285],[57,15503],{"className":15504,"style":1281},[118],[57,15506,15508,15511,15514,15517,15557],{"className":15507},[102],[57,15509],{"className":15510,"style":107},[106],[57,15512,8729],{"className":15513,"style":114},[111,112,113],[57,15515],{"className":15516,"style":119},[118],[57,15518,15520,15523],{"className":15519},[123],[57,15521,3941],{"className":15522},[123,127],[57,15524,15526],{"className":15525},[132],[57,15527,15529,15549],{"className":15528},[136,137],[57,15530,15532,15546],{"className":15531},[141],[57,15533,15535],{"className":15534,"style":146},[145],[57,15536,15537,15540],{"style":550},[57,15538],{"className":15539,"style":154},[153],[57,15541,15543],{"className":15542},[158,159,160,161],[57,15544,88],{"className":15545},[123,127,161],[57,15547,169],{"className":15548},[168],[57,15550,15552],{"className":15551},[141],[57,15553,15555],{"className":15554,"style":176},[145],[57,15556],{},[57,15558,15560,15563],{"className":15559},[123],[57,15561,3000],{"className":15562},[123,127],[57,15564,15566],{"className":15565},[132],[57,15567,15569,15589],{"className":15568},[136,137],[57,15570,15572,15586],{"className":15571},[141],[57,15573,15575],{"className":15574,"style":146},[145],[57,15576,15577,15580],{"style":550},[57,15578],{"className":15579,"style":154},[153],[57,15581,15583],{"className":15582},[158,159,160,161],[57,15584,88],{"className":15585},[123,127,161],[57,15587,169],{"className":15588},[168],[57,15590,15592],{"className":15591},[141],[57,15593,15595],{"className":15594,"style":176},[145],[57,15596],{}," 就能拼出任何你要的结果。",{"title":15599,"searchDepth":15600,"depth":15600,"links":15601},"",2,[15602,15604,15605,15606],{"id":40,"depth":15603,"text":41},3,{"id":2327,"depth":15603,"text":2328},{"id":13257,"depth":15603,"text":13258},{"id":14995,"depth":15603,"text":14995},"md",{"order":15609},8,true,"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fringdirectprod",{"title":5,"description":16},"source\u002Fmath\u002Fabstract_algebra\u002Fringdirectprod","6bDlENKflFZpDVo48Tow-JKqhLL9N-4tZKSoXLp7Z8s",null,1784032893882]