{"id":23169,"date":"2020-10-22T00:03:37","date_gmt":"2020-10-21T16:03:37","guid":{"rendered":"https:\/newsletter.sinica.edu.tw\/?p=23169"},"modified":"2021-08-26T08:50:37","modified_gmt":"2021-08-26T00:50:37","slug":"%e3%80%90%e5%b0%88%e6%ac%84%e3%80%91%e5%be%9e%e4%bb%a3%e6%95%b8%e5%8f%b2%e5%88%b0%e7%95%b6%e4%bb%a3%e8%a1%a8%e7%8f%be%e7%90%86%e8%ab%96","status":"publish","type":"post","link":"https:\/newsletter.sinica.edu.tw\/23169\/","title":{"rendered":"\u3010\u5c08\u6b04\u3011\u5f9e\u4ee3\u6578\u53f2\u5230\u7576\u4ee3\u8868\u73fe\u7406\u8ad6"},"content":{"rendered":"

\u4e00\u3001
\n\u4ee3\u6578\u5b78\u662f\u6578\u5b78\u91cd\u8981\u7684\u5206\u652f\uff0c\u5f9e\u570b\u6c11\u7fa9\u52d9\u6559\u80b2\u958b\u59cb\uff0c\u5b78\u751f\u5c31\u8981\u5b78\u7fd2\u300c\u4ee3\u6578\u300d\uff0c\u4e00\u908a\u8a2d\u672a\u77e5\u6578\uff0c\u4e00\u908a\u601d\u8003\u70ba\u4ec0\u9ebc\u9019\u4e16\u754c\u4e0a\u6709\u90a3\u9ebc\u591a\u4eba\u559c\u6b61\u628a\u96de\u548c\u5154\u5b50\u653e\u5728\u540c\u4e00\u500b\u7c60\u5b50\u88e1\u3002\u6709\u8da3\u7684\u662f\uff0c\u4e0d\u540c\u7684\u7fa4\u9ad4\u5c0d\u65bc\u4ee3\u6578\u662f\u4ec0\u9ebc\u6709\u8457\u622a\u7136\u4e0d\u540c\u7684\u770b\u6cd5\u3002\u4e2d\u5b78\u751f\u89ba\u5f97\u4ee3\u6578\u5c31\u662f\u89e3 x\uff0c\u5927\u5b78\u751f\u89ba\u5f97\u4ee3\u6578\u5c31\u662f\u7fa4\u74b0\u9ad4\uff0c\u800c\u6578\u5b78\u5bb6\u5247\u901a\u5e38\u6c92\u8fa6\u6cd5\u5f88\u7c21\u6f54\u7684\u544a\u8a34\u4f60\u7b54\u6848\uff0c\u56e0\u70ba\u6578\u5b78\u5bb6\u77e5\u9053\u7684\u592a\u591a\u4e86\u3002\u672c\u6587\u4e2d\u7b46\u8005\u5617\u8a66\u5f9e\u6b77\u53f2\u6f14\u9032\u51fa\u767c\uff0c\u804a\u804a\u4e0d\u540c\u6642\u4ee3\u4ee3\u6578\u4e00\u8a5e\u4ee3\u8868\u7684\u610f\u7fa9\uff0c\u6700\u5f8c\u7c21\u55ae\u4ecb\u7d39\u8fd1\u4e16\u4ee3\u6578\u4e2d\u91cd\u8981\u7684\u5b50\u9818\u57df\u2014\u2014\u8868\u73fe\u7406\u8ad6\uff08representation theory\uff09\u3002<\/p>\n

\u5728\u5341\u4e5d\u4e16\u7d00\u4e4b\u524d\uff0c\u4ee3\u6578\u5b78\u6db5\u84cb\u7684\u7bc4\u570d\u90fd\u53ea\u662f\u5c0b\u627e\u56db\u6b21\u4ee5\u4e0b\u7684\u591a\u9805\u5f0f\u65b9\u7a0b\u5f0f\u7684\u89e3\u3002\u96d6\u7136\u5f9e\u897f\u5143\u524d\u4e00\u5343\u4e03\u767e\u5e74\u7684\u5df4\u6bd4\u502b\u4eba\u5011\u5c31\u958b\u59cb\u7b97\u4ee3\u6578\uff0c\u6703\u89e3\u4e8c\u6b21\u591a\u9805\u5f0f\u4e86\uff0c\u4ee3\u6578\u6210\u70ba\u4e00\u9580\u5b78\u554f\u8981\u59cb\u65bc\u963f\u62c9\u4f2f\u4e16\u754c\u7684\u9a55\u50b2\u2014\u2014\u6578\u5b78\u5bb6\u82b1\u62c9\u5b50\u7c73\uff08al-Khwarizmi\uff09\u3002\u82b1\u62c9\u5b50\u7c73\u5728824\u5e74\u5beb\u4e86\u672c\u5c08\u66f8\u300a\u79fb\u9805\u8207\u6d88\u53bb\u4e4b\u8a08\u7b97\u7e3d\u6210\u300b(al-mukhtasar fi hisab al-jabr wa al-muqabala<\/em>\uff09\uff0c\u5176\u4e2d\u79fb\u9805 al-jabr \u4e00\u8a5e\u7684\u62c9\u4e01\u7ffb\u8b6f algebra \u5c31\u662f\u4ee3\u6578\u4e00\u8a5e\u7684\u8d77\u6e90\u3002<\/p>\n

\u4e0a\u53e4\u6642\u671f\u7684\u4ee3\u6578\uff0c\u5c31\u662f\u79fb\u9805\u3002<\/em><\/strong><\/p>\n

\u4e8c\u3001
\n\u5230\u4e86\u6587\u85dd\u5fa9\u8208\u6642\u671f\uff0c\u6578\u5b78\u793e\u7fa4\u6709\u516c\u958b\u6311\u6230\u7684\u98a8\u6c23\uff0c\u5169\u500b\u6578\u5b78\u5bb6\u6703\u70ba\u4e86\u540d\u8b7d\u4e92\u76f8\u5411\u5c0d\u65b9\u63d0\u51fa\u554f\u984c\u6c7a\u9b25\uff0c\u4e5f\u5c31\u662f\u6bd4\u8ab0\u80fd\u89e3\u65b9\u7a0b\u5f0f\u89e3\u5f97\u53c8\u5feb\u53c8\u597d\u30021535 \u5e74\uff0c\u7fa9\u5927\u5229\u6578\u5b78\u5bb6\u5854\u5854\u5229\u4e9e\uff08Tartaglia\uff09\u548c\u8cbb\u723e\uff08Fior\uff09\u5c55\u958b\u4e86\u4e00\u5834\u8457\u540d\u7684\u6c7a\u9b25\uff0c\u4ed6\u5011\u4e92\u76f8\u51fa\u4e86\u4e09\u5341\u984c\u4e09\u6b21\u65b9\u7a0b\u5f0f\u7d66\u5c0d\u65b9\u89e3\u3002\u8cbb\u723e\u624b\u4e2d\u63e1\u6709\u4ed6\u7684\u8001\u5e2b\u5fb7\u8cbb\u7f85\uff08del Ferro\uff09\u89aa\u50b3\u7684\u7955\u65b9\uff0c\u53ef\u4ee5\u627e\u51fa\u5f62\u5982 x^3+bx=c \u7684\u65b9\u7a0b\u5f0f\u89e3\uff1b\u4f46\u662f\u5854\u5854\u5229\u4e9e\u9664\u4e86\u8cbb\u723e\u6240\u77e5\u7684\u5f62\u5f0f\u4ee5\u5916\uff0c\u4e5f\u540c\u6642\u80fd\u89e3\u6c7a\u5f62\u5982 x^3+\u3016bx\u3017^2=c \u7684\u65b9\u7a0b\u5f0f\u3002\u6700\u5f8c\u5854\u5854\u5229\u4e9e\u5728\u6bd4\u8cfd\u958b\u59cb\u7684\u5169\u5c0f\u6642\u5c31\u628a\u8cbb\u723e\u51fa\u7d66\u4ed6\u7684\u4e09\u5341\u984c\u76e1\u6578\u89e3\u51fa\uff0c\u4ee5 30\u6bd40\u75db\u5bb0\u4e86\u534a\u6876\u6c34\u7684\u8cbb\u723e\u3002<\/p>\n

\u9019\u500b\u7d50\u679c\u9707\u9a5a\u4e86\u7fa9\u5927\u5229\u6578\u5b78\u754c\uff0c\u7576\u6642\u4eba\u5011\u56e0\u70ba\u7f3a\u5c11\u8ca0\u6578\u8207\u8907\u6578\u7684\u6982\u5ff5\uff0c\u666e\u904d\u76f8\u4fe1\u4e09\u6b21\u65b9\u7a0b\u5f0f\u662f\u4e0d\u53ef\u89e3\u7684\u3002\u5728\u7c73\u862d\u4efb\u6559\u7684\u5361\u7576\uff08Cardan\uff09\u4fbf\u662f\u5982\u6b64\uff0c\u4ed6\u82e6\u82e6\u61c7\u6c42\u5854\u5854\u5229\u4e9e\u50b3\u6388\u89e3\u6cd5\uff0c\u751a\u81f3\u63d0\u8b70\u8981\u4ecb\u7d39\u7c73\u862d\u5927\u5b98\u7d66\u5854\u5854\u5229\u4e9e\uff0c\u8b93\u4ed6\u6216\u8a31\u80fd\u5f97\u5230\u66f4\u597d\u7684\u5de5\u4f5c\u3002\u5854\u5854\u5229\u4e9e\u6700\u5f8c\u8981\u5361\u7576\u767c\u8a93\u4e0d\u628a\u89e3\u6cd5\u5916\u50b3\uff0c\u624d\u52c9\u5f37\u5c07\u89e3\u6cd5\u7528\u96b1\u6666\u7684\u8a69\u53e5\u5beb\u7d66\u5361\u7576\u3002\u4e0d\u4e45\u4ee5\u5f8c\uff0c\u5361\u7576\u5f9e\u5fb7\u8cbb\u7f85\u7684\u53e6\u4e00\u500b\u5f1f\u5b50\u53e3\u4e2d\u4e5f\u5b78\u5230\u4e86\u5fb7\u8cbb\u7f85\u6d3e\u7684\u89e3\u6cd5\u3002\u5361\u7576\u548c\u81ea\u5df1\u7684\u5f1f\u5b50\u6cd5\u62c9\u5229\uff08Ferrari\uff09\u5c07\u4e09\u6b21\u7684\u89e3\u6cd5\u63a8\u5ee3\u767c\u5c55\u51fa\u56db\u6b21\u65b9\u7a0b\u5f0f\u7684\u89e3\u6cd5\uff0c\u767c\u8868\u5728\u4ed6 1545 \u5e74\u7684\u66f8\u300a\u5927\u8853<\/em>\u300b\uff08Artis magnae<\/em>\uff09\u4e2d\u3002\u5361\u7576\u89ba\u5f97\u81ea\u5df1\u4e26\u4e0d\u662f\u767c\u8868\u5854\u5854\u5229\u4e9e\u7684\u5de5\u4f5c\uff0c\u56e0\u6b64\u6c92\u6709\u9055\u80cc\u7576\u521d\u7684\u8a93\u8a00\u3002\u5854\u5854\u5229\u4e9e\u5c0d\u6b64\u6012\u4e0d\u53ef\u6291\uff0c\u66f8\u4fe1\u5f80\u4f86\u5492\u7f75\u62a8\u64ca\u6578\u5e74\uff0c\u6700\u5f8c\u5728 1548 \u5e74\u7d04\u5b9a\u4e86\u8207\u6cd5\u62c9\u5229\u7684\u6c7a\u9b25\u3002\u7576\u5929\u7fa4\u773e\u805a\u96c6\u5728\u7c73\u862d\u7684\u4e00\u500b\u6559\u5802\uff0c\u4e0b\u5f9e\u8ca9\u592b\u8d70\u5352\u4e0a\u81f3\u7c73\u862d\u5e02\u9577\u90fd\u4f86\u770b\u71b1\u9b27\u3002\u53ea\u662f\u548c\u524d\u4e00\u6b21\u6c7a\u9b25\u4e0d\u540c\u7684\u662f\uff0c\u5854\u5854\u5229\u4e9e\u96d6\u7136\u638c\u63e1\u4e86\u4e09\u6b21\u65b9\u7a0b\uff0c\u4f46\u662f\u5c0d\u65bc\u56db\u6b21\u65b9\u7a0b\u537b\u529b\u6709\u672a\u902e\uff0c\u5728\u6c7a\u9b25\u7b2c\u4e00\u5929\u7684\u665a\u4e0a\u8d81\u591c\u9041\u9003\uff0c\u8b93\u6cd5\u62c9\u5229\u53d6\u5f97\u9019\u5834\u6c7a\u9b25\u7684\u52dd\u5229\u3002<\/p>\n

\u6587\u85dd\u5fa9\u8208\u6642\u7684\u4ee3\u6578\uff0c\u5c31\u662f\u6c7a\u9b25\u3002<\/strong><\/em><\/p>\n

\u4e09\u3001
\n\u5230\u4e86 1770 \u5e74\uff0c\u7fa9\u88d4\u6cd5\u7c4d\u6578\u5b78\u5bb6\u62c9\u683c\u6717\u65e5\uff08Lagrange\uff09\u5c0d\u56db\u6b21\u4ee5\u4e0b\u7684\u65b9\u7a0b\u5f0f\u820a\u6709\u89e3\u6cd5\u505a\u4e86\u7a81\u7834\u6027\u7684\u5206\u6790: \u8981\u89e3\u4e00\u500b n \u6b21\u65b9\u7a0b\u5f0f f(x)=0\uff0c\u53ef\u4ee5\u5f9e\u4e00\u500b\u8f14\u52a9\u7684\u65b9\u7a0b\u5f0f R(x)=0 \u958b\u59cb\uff0cR(x) \u7684\u6839\u5c0d\u61c9\u5230 f(x) \u7684 n \u500b\u6839\u7684\u7f6e\u63db\uff08permutation\uff09\uff0c\u4e5f\u5c31\u662f\u8aaa R(x) \u70ba\u4e00\u500b n! \u6b21\u591a\u9805\u5f0f\u3002<\/p>\n

\u4f8b\uff1a\u7576 n=3\u7684\u6642\u5019\uff0c\u7e3d\u5171\u6709 3! = 6 \u500b\u7f6e\u63db\uff1a 123,132,213,231,312,321\u3002<\/p>\n

\u62c9\u683c\u6717\u65e5\u8b49\u660e\u4e86\u5982\u679c\u8f14\u52a9\u65b9\u7a0b\u5f0f\u80fd\u89e3\uff0c\u5247\u539f\u65b9\u7a0b\u5f0f\u53ef\u89e3\u3002\u5728 f(x) \u6b21\u6578\u70ba 3 \u7684\u60c5\u6cc1\u4e0b\uff0cR(x) \u525b\u597d\u662f\u500b x^3 \u7684\u4e8c\u6b21\u591a\u9805\u5f0f\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u662f\u500b\u507d\u88dd\u6210 3! = 6 \u6b21\u591a\u9805\u5f0f\u7684 2 \u6b21\u591a\u9805\u5f0f\uff1b\u5728 f(x) \u6b21\u6578\u70ba 4 \u7684\u60c5\u6cc1\u4e0b\uff0c R(x) \u525b\u597d\u662f\u500b\u507d\u88dd\u6210 4! = 24 \u6b21\u591a\u9805\u5f0f\u7684 3 \u6b21\u591a\u9805\u5f0f\uff0c\u56e0\u6b64\u6211\u5011\u4f3c\u4e4e\u7e3d\u80fd\u5c07\u65b9\u7a0b\u5f0f\u964d\u6b21\u89e3\u6c7a\u3002\u4f46\u662f\u5728 f(x) \u6b21\u6578\u70ba 5 \u7684\u60c5\u6cc1\u4e0b\uff0c R(x) \u6210\u4e86\u4e00\u500b\u507d\u88dd\u6210 5! = 120 \u6b21\u591a\u9805\u5f0f\u7684 24 \u6b21\u591a\u9805\u5f0f\uff0c\u6b21\u6578\u4e0d\u6e1b\u53cd\u589e\u3002\u62c9\u683c\u6717\u65e5\u56e0\u6b64\u731c\u6e2c\u4e94\u6b21\u4ee5\u4e0a\u7684\u65b9\u7a0b\u5f0f\u4e0d\u4fdd\u8b49\u53ef\u89e3\uff0c\u6700\u5f8c\u9019\u500b\u5b9a\u7406\u7531\u632a\u5a01\u6578\u5b78\u5bb6\u963f\u8c9d\u723e\uff08Abel\uff09\u65bc 1824\u5e74\u8b49\u660e\u3002\u9806\u5e36\u4e00\u63d0\uff0c\u8afe\u8c9d\u723e\u734e\u7684\u6578\u5b78\u7248\u672c\u00ac\u5c31\u53eb\u505a\u963f\u8c9d\u723e\u734e\uff0c\u7531\u632a\u5a01\u653f\u5e9c\u51fa\u8cc7\u9010\u5e74\u9812\u767c\uff0c\u548c\u53ea\u91dd\u5c0d 40\u6b72\u4ee5\u4e0b\u7684\u8cbb\u723e\u8332\u734e\uff08Fields Medal\uff09\u4e0d\u540c\uff0c\u963f\u8c9d\u723e\u734e\u6c92\u6709\u5e74\u9f61\u9650\u5236\u30021831\u5e74\uff0c\u6cd5\u570b\u6578\u5b78\u5bb6\u4f3d\u7f85\u74e6\uff08Galois\uff09\u5960\u57fa\u5728\u62c9\u683c\u6717\u65e5\u548c\u963f\u8c9d\u723e\u7684\u5de5\u4f5c\u4e4b\u4e0a\uff0c\u7d66\u51fa\u4e86\u65b9\u7a0b\u5f0f\u53ef\u89e3\u7684\u5145\u8981\u689d\u4ef6\uff0c\u7528\u73fe\u4ee3\u7684\u8a9e\u8a00\u4f86\u8aaa\uff0c\u5224\u65b7\u65b9\u7a0b\u5f0f\u662f\u5426\u53ef\u89e3\u7b49\u50f9\u65bc\u4e86\u89e3\u7f6e\u63db\u7fa4\uff08permutation group\uff09\u7684\u5167\u5728\u7d50\u69cb\u3002\u53ef\u60dc\u7684\u662f\uff0c\u4f3d\u7f85\u74e6\u70ba\u4e86\u611b\u60c5\u6b7b\u65bc\u771f\u5200\u771f\u69cd\u7684\u6c7a\u9b25\u4e4b\u4e2d\uff0c\u8981\u662f\u4f3d\u7f85\u74e6\u80fd\u7528\u4ee3\u6578\u6c7a\u9b25\u5c31\u597d\u4e86\u3002<\/p>\n

\u5341\u4e5d\u4e16\u7d00\u521d\u671f\u7684\u4ee3\u6578\uff0c\u5c31\u662f\u6839\u7684\u7f6e\u63db\u3002<\/strong><\/em><\/p>\n

\u56db\u3001
\n\u81f3\u5341\u4e5d\u4e16\u7d00\u5f8c\u534a\uff0c\u4ee3\u6578\u7684\u767c\u5c55\u958b\u59cb\u767c\u6563\u30021870\u5e74\uff0c\u632a\u5a01\u6578\u5b78\u5bb6\u674e\uff08Lie\uff09\u548c\u666e\u9b6f\u58eb\u6578\u5b78\u5bb6\u514b\u840a\u6069\uff08Klein\uff09\u540c\u6642\u5230\u5df4\u9ece\u8a2a\u554f\uff0c\u505a\u4e86\u5169\u500b\u6708\u7684\u9130\u5c45\u8207\u540c\u4e8b\u3002\u64c5\u9577\u4ee3\u6578\u7684\u514b\u840a\u6069\u559c\u6b61\u7814\u7a76\u7279\u4f8b\uff1b\u64c5\u9577\u5206\u6790\u7684\u674e\u559c\u6b61\u601d\u8003\u6700\u4e00\u822c\u7684\u60c5\u6cc1\u3002\u5408\u4f5c\u5230\u4e00\u534a\uff0c\u666e\u6cd5\u6230\u722d\u7206\u767c\uff0c\u514b\u840a\u6069\u4e0d\u5f97\u4e0d\u8fd4\u56de\u666e\u9b6f\u58eb\uff0c\u800c\u674e\u904e\u4e00\u9663\u5b50\u8a08\u756b\u907f\u96e3\u5230\u7fa9\u5927\u5229\uff0c\u537b\u5728\u6953\u4e39\u767d\u9732\u88ab\u7576\u4f5c\u666e\u9b6f\u58eb\u9593\u8adc\u902e\u6355\uff0c\u674e\u96a8\u8eab\u651c\u5e36\u7684\u6578\u5b78\u624b\u7a3f\u9084\u88ab\u7576\u4f5c\u52a0\u5bc6\u5f8c\u7684\u6a5f\u5bc6\u60c5\u5831\u9ad8\u898f\u683c\u5c0d\u5f85\u3002\u9084\u597d\u6709\u719f\u8b58\u7684\u6cd5\u570b\u6578\u5b78\u5bb6\u5c07\u674e\u4fdd\u51fa\u4f86\uff0c\u4e0d\u7136\u6578\u5b78\u7684\u767c\u5c55\u53ef\u80fd\u6703\u88ab\u62d6\u7d2f\u6578\u5341\u5e74\u3002 1871 \u5e74\uff0c\u674e\u548c\u514b\u840a\u6069\u767c\u8868\u4e86\u8b8a\u5f62\u7fa4\uff08transformation group\uff09\u7684\u6982\u5ff5\u4f86\u7814\u7a76\u7269\u9ad4\u5728\u7a7a\u9593\u4e2d\u7684\u904b\u52d5\u8207\u5e7e\u4f55\uff0c\u53ef\u4ee5\u6db5\u84cb\u9023\u7e8c\u6216\u96e2\u6563\u7684\u60c5\u6cc1\u3002\u5169\u4eba\u7684\u7814\u7a76\u8208\u8da3\u4e4b\u5f8c\u5f80\u4e0d\u540c\u7684\u65b9\u5411\u5206\u6b67\uff1a\u514b\u840a\u6069\u95dc\u5fc3\u96e2\u6563\u8b8a\u5f62\uff0c\u767c\u5c55\u4e86\u7f6e\u63db\u7fa4\u4ee5\u5916\u7684\u6709\u9650\u7fa4\u8ad6\uff1b\u674e\u95dc\u5fc3\u9023\u7e8c\u8b8a\u5f62\uff0c\u628a\u65b9\u7a0b\u5f0f\u8207\u7f6e\u63db\u7fa4\u4e4b\u9593\u7684\u9023\u7d50\u63a8\u5ee3\uff0c\u5efa\u7acb\u4e00\u985e\u7279\u6b8a\u7684\u7121\u9650\u7fa4\u4f86\u8655\u7406\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\uff0c\u9019\u985e\u7121\u9650\u7fa4\u73fe\u5728\u7a31\u70ba\u674e\u7fa4\uff08Lie group\uff09\u3002<\/p>\n

\u674e\u7fa4\u7406\u8ad6\u7684\u4e00\u500b\u91cd\u9ede\u662f\uff0c\u8207\u5176\u7814\u7a76\u674e\u7fa4\u672c\u8eab\uff0c\u4e0d\u5982\u7814\u7a76\u674e\u7fa4\u6240\u8a98\u751f\u7684\u5d84\u65b0\u7d50\u69cb\uff0c\u674e\u5c07\u9019\u500b\u65b0\u7d50\u69cb\u7a31\u70ba\u300c\u7121\u7aae\u5c0f\u7fa4\u300d\uff0c\u5728\u73fe\u4eca\u7684\u8a9e\u8a00\u4e2d\u7a31\u4f5c\u674e\u4ee3\u6578\uff08Lie algebra\uff09\u3002\u7d04\u7565\u4f86\u8aaa\uff0c\u674e\u4ee3\u6578\u662f\u500b\u5e36\u6709\u984d\u5916\u64cd\u4f5c\u7684\u5411\u91cf\u7a7a\u9593\uff0c\u88e1\u9762\u7684\u5411\u91cf\u53ef\u4ee5\u76f8\u52a0\u3001\u53ef\u4ee5\u5ef6\u9577\u3001\u4f46\u662f\u4e0d\u80fd\u76f8\u4e58\u3002\u674e\u4ee3\u6578\u81ea\u5e36\u7684\u984d\u5916\u64cd\u4f5c\u53eb\u505a\u674e\u62ec\u865f\uff08Lie bracket\uff09\uff0c\u53ef\u4ee5\u7528\u4f86\u7522\u751f\u65b0\u7684\u5411\u91cf\uff0c\u4f8b\u5982\u8aaa\u6211\u5011\u53ef\u4ee5\u5c07\u5411\u91cf v \u548c w \u900f\u904e\u674e\u62ec\u865f\u5305\u8d77\u4f86\u7522\u751f\u4e00\u500b\u65b0\u7684\u5411\u91cf\uff0c\u8a18\u505a [v,w] \u3002\u4e00\u822c\u4f86\u8aaa\u674e\u62ec\u865f\u5fc5\u9808\u6eff\u8db3\u5305\u542b\u8cc8\u53ef\u6bd4\uff08Jacobi\uff09\u6046\u7b49\u5f0f\u5728\u5167\u7684\u4e00\u4e9b\u689d\u4ef6\uff0c\u4f46\u662f\u5982\u679c\u6211\u5011\u8a2d\u5b9a\u8b93\u6240\u6709\u7684 [v,w] \u90fd\u7b49\u65bc\u96f6\uff0c\u6240\u6709\u9808\u6aa2\u67e5\u7684\u689d\u4ef6\u90fd\u6703\u6210\u7acb\uff0c\u4e5f\u5c31\u5f62\u6210\u6700\u7c21\u55ae\u7684\u674e\u4ee3\u6578\u2014\u2014\u963f\u8c9d\u723e\u674e\u4ee3\u6578\u3002\u4e00\u822c\u4f86\u8aaa [v,w] \u7576\u7136\u53ef\u4ee5\u662f\u975e\u96f6\u5411\u91cf\uff0c\u9019\u6a23\u5b9a\u7fa9\u51fa\u4f86\u7684\u674e\u4ee3\u6578\u8907\u96dc\u7684\u591a\uff0c\u5206\u985e\u4e26\u523b\u5283\u674e\u4ee3\u6578\u4fbf\u6210\u70ba\u6578\u5b78\u5bb6\u95dc\u5fc3\u7684\u554f\u984c\u3002\u8907\u6578\u4e0a\u7684\u6709\u9650\u7dad\u674e\u4ee3\u6578\u5206\u985e\u554f\u984c\u7531\u6cd5\u570b\u6578\u5b78\u5bb6\u52a0\u7576\uff08Cartan\uff09\u8207\u5fb7\u570b\u6578\u5b78\u5bb6\u9f4a\u6797\uff08Killing\uff09\u5728 1888-1894\u5b8c\u6210\u3002\u800c\u7121\u9650\u7dad\u674e\u4ee3\u6578\u7684\u5206\u985e\u8981\u5230 1967 \u5e74\u624d\u7531\u4fc4\u570b\u6578\u5b78\u5bb6\u5361\u8332\uff08Kac\uff09\u548c\u52a0\u62ff\u5927\u6578\u5b78\u5bb6\u7a46\u6575\uff08Moody\uff09\u5206\u5225\u7368\u7acb\u5b8c\u6210\u3002<\/p>\n

\u53e6\u4e00\u65b9\u9762\uff0c\u4eba\u5011\u5f9e\u5df4\u6bd4\u502b\u6642\u4ee3\u958b\u59cb\uff0c\u89e3\u4e8c\u6b21\u591a\u9805\u5f0f\u7684\u6642\u5019\u5c31\u9047\u898b\u8907\u6578\u4e86\u3002\u8d85\u904e\u5343\u5e74\u7684\u7121\u8996\u4e5f\u6c92\u6709\u8fa6\u6cd5\u62b9\u6ec5\u4e00\u500b\u6578\u5b78\u6982\u5ff5\uff0c\u751a\u4e4b\uff0c\u7121\u4eba\u770b\u7ba1\u7684\u706b\u82d7\u53ea\u6703\u8d8a\u71d2\u8d8a\u65fa\u30021843\u5e74\u611b\u723e\u862d\u6578\u5b78\u5bb6\u6f22\u7c73\u723e\u9813\uff08Hamilton\uff09\u5c07\u8907\u6578\u7cfb\u770b\u6210\u6240\u6709\u5f62\u5982 a1+bi \u7684\u300c\u6578\u300d\uff0c\u5176\u4e2d a \u548c b \u5fc5\u9808\u662f\u5be6\u6578\uff0c1\u548c i \u662f\u5169\u500b\u9700\u8981\u63cf\u8ff0\u7279\u6b8a\u4e58\u6cd5\u898f\u5247\uff08\u5373 1*1=1,1*i=i*1=i,i*i=-1\uff09\u7684\u300c\u751f\u6210\u5143\u300d\u3002\u6f22\u7c73\u723e\u9813\u7acb\u523b\u5c31\u60f3\u5230\u4ed6\u53ef\u4ee5\u5c07\u8907\u6578\u518d\u63a8\u5ee3\u2014\u2014\u589e\u52a0\u9700\u8981\u5b9a\u7fa9\u7279\u6b8a\u4e58\u6cd5\u898f\u5247\u7684\u751f\u6210\u5143\u500b\u6578\uff0c\u4f86\u5b9a\u7fa9\u6bd4\u8907\u6578\u9084\u8907\u96dc\u7684\u6578\u7cfb\u3002\u6f22\u7c73\u723e\u9813\u6210\u529f\u7684\u5b9a\u7fa9\u4e86\u56db\u5143\u6578\uff08quaternion\uff09\uff0c\u7576\u4e2d\u6240\u6709\u7684\u300c\u6578\u300d\u5f62\u5982 a1+bi+cj+dk\uff0c\u5176\u4e2da,b,c,d \u662f\u5be6\u6578\u3001 1,i,j,k \u662f\u751f\u6210\u5143\u3002\u4ed6\u7684\u540c\u4e8b\u845b\u96f7\u592b\uff08Graves\uff09\u5728\u807d\u5230\u6d88\u606f\u7684\u9694\u5929\u5c31\u4eff\u9020\u56db\u5143\u6578\u8a02\u51fa\u4e86\u516b\u5143\u6578\uff08octonion\uff09\u3002\u9019\u6a23\u914d\u5408\u73fe\u6709\u7684\u6578\u7cfb\u5b9a\u51fa\u984d\u5916\u4e58\u6cd5\u898f\u5247\u4f86\u5b9a\u7fa9\u7684\u7d50\u69cb\uff0c\u5f88\u5feb\u5730\u88ab\u767c\u73fe\u51fa\u73fe\u5728\u5176\u4ed6\u6578\u5b78\u9818\u57df\u4e2d\u3002\u6bd4\u65b9\u8aaa\u6240\u6709\u7684 n\u968e\u77e9\u9663\u7d44\u6210\u7684\u96c6\u5408\uff0c\u7576\u4e2d\u6bcf\u500b\u5143\u7d20\u53ef\u4ee5\u8996\u70ba\u57fa\u790e\u77e9\u9663 E_ij\u7684\u7dda\u6027\u7d44\u5408\uff0c\u800c\u984d\u5916\u7684\u4e58\u6cd5\u898f\u5247\u5c31\u662f
\nE_ij*E_iy=E_iy,E_ij*E_xy=0 (\u82e5\u3000x\u2260j)
\n\u6b64\u985e\u7d50\u69cb\uff0c\u7531\u65bc\u901a\u5e38\u662f\u5f9e\u8907\u6578\u64f4\u5f35\u800c\u6210\uff0c\u7a31\u70ba\u8d85\u8907\u96dc\u6578\u7cfb\uff08hypercomplex number system\uff09\u3002<\/p>\n

1870\u5e74\uff0c\u7f8e\u570b\u6578\u5b78\u5bb6\u76ae\u723e\u65af\uff08Peirce\uff09\u9996\u6b21\u5b9a\u7fa9\u4e86\u7d50\u5408\u4ee3\u6578\uff08associative algebra\uff09\uff0c\u7c21\u7a31\u4ee3\u6578\u3002\u5927\u7565\u4f86\u8aaa\uff0c\u4e00\u500b\u4ee3\u6578\u4e2d\u7684\u5143\u7d20\u4e5f\u662f\u5411\u91cf\uff0c\u6bcf\u500b\u5411\u91cf\u9664\u4e86\u53ef\u4ee5\u5ef6\u9577\u3001\u53ef\u4ee5\u758a\u52a0\u4ee5\u5916\uff0c\u9084\u53ef\u4ee5\u300c\u76f8\u4e58\u300d\u3002\u9019\u88e1\u7684\u4e58\u6cd5\u4e5f\u8981\u984d\u5916\u6307\u5b9a\u4fc2\u6578\u4f86\u63cf\u8ff0\u4e58\u6cd5\u898f\u5247\uff0c\u53ea\u662f\u898f\u5247\u7684\u9650\u5236\u548c\u674e\u4ee3\u6578\u4e0d\u540c\u3002\u76ae\u723e\u65af\u5728\u4ed6\u8d85\u904e\u4e00\u767e\u9801\u7684\u6587\u7ae0\u4e2d\u63cf\u8ff0\u4e86 150 \u7a2e\u4e0d\u540c\u7684\u516d\u7dad\u4ee5\u4e0b\u4ee3\u6578\u7d50\u69cb\uff0c\u4e5f\u5275\u9020\u4e86\u5982\u51aa\u96f6\uff08nilpotent\uff09\u6216\u662f\u51aa\u7b49\uff08idempotent\uff09\u6b64\u985e\u81f3\u4eca\u4ecd\u7136\u88ab\u5ee3\u6cdb\u4f7f\u7528\u7684\u6578\u5b78\u6982\u5ff5\u3002\u5230\u4e861908\u5e74\uff0c\u8607\u683c\u862d\u6578\u5b78\u5bb6\u97cb\u5fb7\u672c\uff08Wedderburn\uff09\u5728\u4ed6\u7684\u535a\u58eb\u8ad6\u6587\u4e2d\u5c0d\u65bc\u8d85\u8907\u96dc\u6578\u7cfb\u5efa\u7acb\u4e86\u901a\u7528\u7684\u4e00\u822c\u6027\u7406\u8ad6\uff0c\u5305\u542b\u7406\u60f3\uff08ideal\uff09\u3001\u534a\u55ae\uff08semisimple\uff09\u3001\u76f4\u7a4d\uff08direct sum\uff09\u3001\u5f35\u91cf\u7a4d\uff08tensor product\uff09\uff0c\u6642\u81f3\u4eca\u65e5\uff0c\u8a31\u591a\u6982\u5ff5\u4ecd\u7136\u5728\u5927\u5b78\u4ee3\u6578\u8ab2\u7a0b\u4e2d\u88ab\u5ee3\u6cdb\u4f7f\u7528\u3002<\/p>\n

\u5341\u4e5d\u4e16\u7d00\u5f8c\u534a\u958b\u59cb\uff0c\u4ee3\u6578\u6210\u70ba\u4e00\u7a2e\u6578\u5b78\u7d50\u69cb\u7684\u540d\u5b57\u3001\u800c\u4ee3\u6578\u5b78\u5c31\u662f\u7814\u7a76\u7d50\u69cb\u3002<\/strong><\/em><\/p>\n

\u4e94\u3001
\n\u8868\u73fe\u4f9d\u8cf4\u65bc\u4e0d\u540c\u7684\u4ee3\u6578\u7d50\u69cb\u4e0a\uff0c\u6211\u5011\u53ef\u4ee5\u5c0d\u4e0d\u540c\u7684\u4ee3\u6578\u7d50\u69cb\u7814\u7a76\u8868\u73fe\u7406\u8ad6\u3002\u6bd4\u65b9\u8aaa\u7fa4\u7684\u8868\u73fe\u7406\u8ad6\u548c\u674e\u4ee3\u6578\u7684\u8868\u73fe\u7406\u8ad6\u5c31\u662f\u4e0d\u540c\u4f46\u662f\u6709\u95dc\u806f\u7684\u7814\u7a76\u4e3b\u984c\u3002\u6253\u500b\u6bd4\u65b9\uff0c\u8868\u73fe\u7406\u8ad6\u4e4b\u65bc\u4ee3\u6578\u7d50\u69cb\u5c31\u50cf\u52d5\u7269\u884c\u70ba\u5b78\u4e4b\u65bc\u91ce\u751f\u52d5\u7269\u3002\u6211\u5011\u6709\u6642\u900f\u904e\u89e3\u5256\u4e0d\u80fd\u5b8c\u5168\u7684\u4e86\u89e3\u52d5\u7269\uff0c\u4f46\u662f\u5f9e\u89c0\u5bdf\u52d5\u7269\u7684\u884c\u70ba\u53ef\u4ee5\u589e\u9032\u6211\u5011\u5c0d\u7260\u7684\u4e86\u89e3\uff0c\u751a\u81f3\u6709\u6642\u5019\uff0c\u4e86\u89e3\u884c\u70ba\u8981\u6bd4\u4e86\u89e3\u52d5\u7269\u672c\u8eab\u66f4\u6709\u610f\u7fa9\u3002<\/p>\n

\u8868\u73fe\u7406\u8ad6\u7684\u524d\u8eab\u662f\u7279\u5fb5\u6a19\uff08character\uff09\u7406\u8ad6\uff0c\u6700\u4e00\u958b\u59cb\u8981\u8ffd\u6eaf\u5230\u5fb7\u570b\u6578\u5b78\u5bb6\u9ad8\u65af\uff08Gauss\uff09\u5728\u6578\u8ad6\u65b9\u9762\u7684\u5de5\u4f5c\uff0c\u6211\u5011\u76ef\u8457\u4ea4\u63db\u7fa4\u770b\uff0c\u770b\u4e0d\u51fa\u592a\u591a\u6709\u610f\u601d\u7684\u7d50\u679c\uff0c\u4f46\u662f\u9019\u7a2e\u7fa4\u7684\u7279\u5fb5\u6a19\u537b\u80fd\u7528\u4f86\u89e3\u7b54\u6578\u8ad6\u4e0a\u7684\u554f\u984c\u2014\u2014\u4ec0\u9ebc\u6a23\u7684\u6574\u6578 n \u80fd\u8868\u793a\u6210\u4e0b\u9762\u7684\u6574\u4fc2\u6578\u4e8c\u6b21\u5f62\u5f0f \uff1f
\nF=\u3016ax\u3017^2+2bxy+\u3016cy\u3017^2
\n\u5c0d\u65bc\u6bcf\u500b\u4e8c\u6b21\u5f62\u5f0f F\uff0c\u9ad8\u65af\u5b9a\u7fa9\u4e86\u5b83\u7684\u884c\u5217\u5f0f\u503c D=b^2-ac \uff0c\u4e26\u8b49\u660e\u4e86\u6240\u6709\u884c\u5217\u5f0f\u503c\u76f8\u540c\u7684\u4e8c\u6b21\u5f62\u5f0f\u6703\u5f62\u6210\u4e00\u500b\u6709\u9650\u4ea4\u63db\u7fa4 G\u3002\u800c G \u7684\u7279\u5fb5\u6a19\u5c31\u662f\u67d0\u4e9b\u51fd\u6578 \u03c7:G\u2192C^*\uff0c\u6eff\u8db3
\n\u03c7(g)\u03c7(h)=\u03c7(gh) \u2200 g,h\u2208G
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 <\/p>\n

\u53c3\u8003\u8cc7\u6599<\/strong>
\n[1] O\u2019Connor, J. J., & Robertson, E. F. (2020). The MacTutor history of mathematics archive<\/em>. http:\/www-history.mcs.st-and.ac.uk<\/a>
\n[2] Kleiner, I. (2007). A history of abstract algebra<\/em>. Springer Science & Business Media.
\n[3] Van der Waerden, B. L. (1985). A history of algebra: From al-Khw\u0101rizm\u012b to Emmy Noether<\/em>. Springer Science & Business Media.<\/p>\n","protected":false},"excerpt":{"rendered":"

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