. "\uD6C4\uB974\uBE44\uCE20\uC758 \uC815\uB9AC(Hurwitz's theorem, -\u5B9A\u7406)\uB294 \uB3C5\uC77C \uC218\uD559\uC790 \uC544\uB3CC\uD504 \uD6C4\uB974\uBE44\uCE20\uC758 \uC774\uB984\uC774 \uBD99\uC740 \uCD94\uC0C1\uB300\uC218\uD559\uC758 \uC815\uB9AC\uB85C, \uD6C4\uB974\uBE44\uCE20\uAC00 1898\uB144 \uC99D\uBA85\uD558\uC600\uB2E4. \uB2E4\uC74C\uACFC \uAC19\uC740 \uB0B4\uC6A9\uC774\uB2E4. \n* \uD56D\uB4F1\uC6D0\uC774 \uC788\uB294 \uC2E4\uC218 \uD639\uC740 \uBCF5\uC18C\uC218 \uC704\uC758 \uB178\uB984\uC774 \uC8FC\uC5B4\uC9C4 \uB85C\uC11C, \uADF8 \uB178\uB984\uC774 \uD56D\uC0C1 \uC744 \uB9CC\uC871\uD558\uB294 \uAC83\uC740 \uD56D\uC0C1 (\uB300\uC218\uB85C\uC11C) \uC2E4\uC218, \uBCF5\uC18C\uC218, \uC0AC\uC6D0\uC218, \uD314\uC6D0\uC218 \uC911 \uD558\uB098\uC640 \uB3D9\uD615\uC774\uB2E4. \uD398\uB974\uB514\uB09C\uD2B8 \uAC8C\uC624\uB974\uD06C \uD504\uB85C\uBCA0\uB2C8\uC6B0\uC2A4\uC5D0 \uC758\uD574 \uC2DC\uC791\uB41C \uC2E4\uC218\uCCB4 \uC704\uC758 \uB098\uB217\uC148 \uB300\uC218\uB97C \uBD84\uB958\uD558\uB294 \uBB38\uC81C\uB294 \uD6C4\uB974\uBE44\uCE20\uAC00 \uC774 \uC815\uB9AC \uB4F1\uC73C\uB85C \uC774\uC5B4\uBC1B\uC544 \uBC1C\uC804\uC2DC\uCF30\uACE0, \uB9C9\uC2A4 \uCD08\uB978\uC774 (alternative ring)\uC758 \uC5F0\uAD6C\uB85C \uC774\uC5D0 \uB300\uD55C \uC77C\uBC18\uC801\uC778 \uACB0\uACFC\uB97C \uC5BB\uC5C8\uB2E4. \uC544\uB3CC\uD504 \uD6C4\uB974\uBE44\uCE20\uB294 \uC774 \uC815\uB9AC\uB97C \uC989\uC2DC \uC751\uC6A9\uD558\uC5EC , \uC624\uC77C\uB7EC\uC758 \uB124 \uC81C\uACF1\uC218 \uD56D\uB4F1\uC2DD\uC774\uB098 \uB370\uAC90\uC758 \uC5EC\uB35F \uC81C\uACF1\uC218 \uD56D\uB4F1\uC2DD\uACFC \uAC19\uC740 \uD56D\uB4F1\uC2DD\uC740 \uBBF8\uC9C0\uC218\uAC00 \uC5EC\uB35F \uAC1C\uBCF4\uB2E4 \uB354 \uB9CE\uC740 \uACBD\uC6B0\uC5D0 \uB300\uD574\uC11C\uB294 \uC131\uB9BD\uD560 \uC218 \uC5C6\uB2E4\uB294 \uC0AC\uC2E4\uC744 \uC99D\uBA85\uD558\uAE30\uB3C4 \uD588\uB2E4."@ko . "\u5728\u4EE3\u6570\u5B66\u4E2D\uFF0C\u80E1\u5C14\u7EF4\u5179\u5B9A\u7406\uFF08\u53C8\u540D\u201C1,2,4,8\u5B9A\u7406\u201D\uFF09\u662F\u4EE5\u57281898\u5E74\u8BC1\u660E\u5B83\u7684\u963F\u9053\u592B\u00B7\u80E1\u5C14\u7EF4\u5179\u547D\u540D\u3002\u8BE5\u5B9A\u7406\u8868\u660E\uFF1A\u4EFB\u4F55\u5E26\u6709\u5355\u4F4D\u5143\u7684\u8CE6\u7BC4\u53EF\u9664\u4EE3\u6578\u540C\u6784\u4E8E\u4EE5\u4E0B\u56DB\u4E2A\u4EE3\u6570\u4E4B\u4E00\uFF1AR,C,H\u548CO\uFF0C\u5206\u522B\u4EE3\u8868\u5B9E\u6570\u3001\u590D\u6570\u3001\u56DB\u5143\u6570\u548C\u516B\u5143\u6570\u3002\u5BF9\u5B9E\u8CE6\u7BC4\u53EF\u9664\u4EE3\u6578\u7684\u5206\u7C7B\u59CB\u4E8E\u5F17\u6D1B\u6BD4\u7EBD\u65AF \uFF0C\u53D1\u626C\u4E8E\u80E1\u5C14\u7EF4\u5179\uFF0C\u7531\u4F50\u6069\u6574\u7406\u4E3A\u4E00\u822C\u5F62\u5F0F\u3002\u4E00\u4E2A\u7B80\u77ED\u7684\u5386\u53F2\u6458\u8981\u53EF\u89C1Badger\u3002 \u5B8C\u6574\u7684\u8BC1\u660E\u80FD\u5728\u51EF\u7279\u548C\u7D22\u6D1B\u591A\u65AF\u5C3C\u79D1\u592B\u6216\u8005\u590F\u76AE\u7F57\u5904\u627E\u5230\u3002\u4E00\u4E2A\u57FA\u672C\u7684\u60F3\u6CD5\u662F\uFF0C\u5982\u679C\u4E00\u4E2A\u4EE3\u6570A\u662F\u6210\u6B63\u6BD4\u4E8E1\u7684\uFF0C\u90A3\u4E48\u5B83\u540C\u6784\u4E8E\u5B9E\u6570\u3002\u5426\u5219\uFF0C\u6211\u4EEC\u4F7F\u7528\u51EF\u83B1-\u8FEA\u514B\u68EE\u7ED3\u6784\u6269\u5C55\u5B50\u4EE3\u6570\u4EE5\u540C\u6784\u4E8E1\uFF0C\u5E76\u5F15\u5165\u4E00\u4E2A\u5411\u91CF\u6B63\u4EA4\u4E8E1\u3002\u6B64\u5B50\u4EE3\u6570\u662F\u540C\u6784\u4E8E\u590D\u6570\u7684\u3002\u5982\u679C\u5B83\u4E0D\u662FA\u7684\u5168\u4F53\uFF0C\u90A3\u4E48\u6211\u4EEC\u518D\u6B21\u4F7F\u7528\u51EF\u83B1-\u8FEA\u514B\u68EE\u7ED3\u6784\u548C\u53E6\u4E00\u4E2A\u4E0E\u590D\u6570\u6B63\u4EA4\u7684\u5411\u91CF\uFF0C\u5F97\u5230\u4E00\u4E2A\u4E0E\u56DB\u5143\u6570\u540C\u6784\u7684\u5B50\u4EE3\u6570\u3002\u5982\u679C\u8FD9\u8FD8\u4E0D\u662F\u4E0D\u662FA\u7684\u5168\u4F53\uFF0C\u6211\u4EEC\u91CD\u590D\u4EE5\u4E0A\u884C\u4E3A\u4E00\u6B21\uFF0C\u5E76\u5F97\u5230\u540C\u6784\u4E8E\uFF08\u6216\u516B\u5143\u6570\uFF09\u7684\u5B50\u4EE3\u6570\u3002\u6211\u4EEC\u73B0\u5728\u6709\u4E00\u4E2A\u5B9A\u7406\uFF0C\u8BF4\u7684\u662F\u6BCF\u4E00\u4E2A\u5305\u542B1\u800C\u53C8\u4E0D\u662FA\u81EA\u8EAB\u7684\u5B50\u4EE3\u6570\u662F\u7ED3\u5408\u7684\u3002\u51EF\u83B1\u6570\u4E0D\u662F\u7ED3\u5408\u7684\uFF0C\u56E0\u6B64\u5FC5\u987B\u4E3AA\u3002 \u80E1\u5C14\u7EF4\u5179\u5B9A\u7406\u4E5F\u53EF\u4EE5\u7528\u4E8E\u8BC1\u660En\u4E2A\u5E73\u65B9\u548C\u4E0En\u4E2A\u5E73\u65B9\u548C\u7684\u79EF\u4ECD\u53EF\u4EE5\u5199\u6210n\u4E2A\u5E73\u65B9\u548C\u4EC5\u5F53n\u4E3A1,2,4\u6216\u80058\u65F6\u3002"@zh . . . . . . . . . . . . . "\u5728\u4EE3\u6570\u5B66\u4E2D\uFF0C\u80E1\u5C14\u7EF4\u5179\u5B9A\u7406\uFF08\u53C8\u540D\u201C1,2,4,8\u5B9A\u7406\u201D\uFF09\u662F\u4EE5\u57281898\u5E74\u8BC1\u660E\u5B83\u7684\u963F\u9053\u592B\u00B7\u80E1\u5C14\u7EF4\u5179\u547D\u540D\u3002\u8BE5\u5B9A\u7406\u8868\u660E\uFF1A\u4EFB\u4F55\u5E26\u6709\u5355\u4F4D\u5143\u7684\u8CE6\u7BC4\u53EF\u9664\u4EE3\u6578\u540C\u6784\u4E8E\u4EE5\u4E0B\u56DB\u4E2A\u4EE3\u6570\u4E4B\u4E00\uFF1AR,C,H\u548CO\uFF0C\u5206\u522B\u4EE3\u8868\u5B9E\u6570\u3001\u590D\u6570\u3001\u56DB\u5143\u6570\u548C\u516B\u5143\u6570\u3002\u5BF9\u5B9E\u8CE6\u7BC4\u53EF\u9664\u4EE3\u6578\u7684\u5206\u7C7B\u59CB\u4E8E\u5F17\u6D1B\u6BD4\u7EBD\u65AF \uFF0C\u53D1\u626C\u4E8E\u80E1\u5C14\u7EF4\u5179\uFF0C\u7531\u4F50\u6069\u6574\u7406\u4E3A\u4E00\u822C\u5F62\u5F0F\u3002\u4E00\u4E2A\u7B80\u77ED\u7684\u5386\u53F2\u6458\u8981\u53EF\u89C1Badger\u3002 \u5B8C\u6574\u7684\u8BC1\u660E\u80FD\u5728\u51EF\u7279\u548C\u7D22\u6D1B\u591A\u65AF\u5C3C\u79D1\u592B\u6216\u8005\u590F\u76AE\u7F57\u5904\u627E\u5230\u3002\u4E00\u4E2A\u57FA\u672C\u7684\u60F3\u6CD5\u662F\uFF0C\u5982\u679C\u4E00\u4E2A\u4EE3\u6570A\u662F\u6210\u6B63\u6BD4\u4E8E1\u7684\uFF0C\u90A3\u4E48\u5B83\u540C\u6784\u4E8E\u5B9E\u6570\u3002\u5426\u5219\uFF0C\u6211\u4EEC\u4F7F\u7528\u51EF\u83B1-\u8FEA\u514B\u68EE\u7ED3\u6784\u6269\u5C55\u5B50\u4EE3\u6570\u4EE5\u540C\u6784\u4E8E1\uFF0C\u5E76\u5F15\u5165\u4E00\u4E2A\u5411\u91CF\u6B63\u4EA4\u4E8E1\u3002\u6B64\u5B50\u4EE3\u6570\u662F\u540C\u6784\u4E8E\u590D\u6570\u7684\u3002\u5982\u679C\u5B83\u4E0D\u662FA\u7684\u5168\u4F53\uFF0C\u90A3\u4E48\u6211\u4EEC\u518D\u6B21\u4F7F\u7528\u51EF\u83B1-\u8FEA\u514B\u68EE\u7ED3\u6784\u548C\u53E6\u4E00\u4E2A\u4E0E\u590D\u6570\u6B63\u4EA4\u7684\u5411\u91CF\uFF0C\u5F97\u5230\u4E00\u4E2A\u4E0E\u56DB\u5143\u6570\u540C\u6784\u7684\u5B50\u4EE3\u6570\u3002\u5982\u679C\u8FD9\u8FD8\u4E0D\u662F\u4E0D\u662FA\u7684\u5168\u4F53\uFF0C\u6211\u4EEC\u91CD\u590D\u4EE5\u4E0A\u884C\u4E3A\u4E00\u6B21\uFF0C\u5E76\u5F97\u5230\u540C\u6784\u4E8E\uFF08\u6216\u516B\u5143\u6570\uFF09\u7684\u5B50\u4EE3\u6570\u3002\u6211\u4EEC\u73B0\u5728\u6709\u4E00\u4E2A\u5B9A\u7406\uFF0C\u8BF4\u7684\u662F\u6BCF\u4E00\u4E2A\u5305\u542B1\u800C\u53C8\u4E0D\u662FA\u81EA\u8EAB\u7684\u5B50\u4EE3\u6570\u662F\u7ED3\u5408\u7684\u3002\u51EF\u83B1\u6570\u4E0D\u662F\u7ED3\u5408\u7684\uFF0C\u56E0\u6B64\u5FC5\u987B\u4E3AA\u3002 \u80E1\u5C14\u7EF4\u5179\u5B9A\u7406\u4E5F\u53EF\u4EE5\u7528\u4E8E\u8BC1\u660En\u4E2A\u5E73\u65B9\u548C\u4E0En\u4E2A\u5E73\u65B9\u548C\u7684\u79EF\u4ECD\u53EF\u4EE5\u5199\u6210n\u4E2A\u5E73\u65B9\u548C\u4EC5\u5F53n\u4E3A1,2,4\u6216\u80058\u65F6\u3002"@zh . . . . . . . . . . . . . . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0413\u0443\u0440\u0432\u0438\u0446\u0430 \u043E \u043D\u043E\u0440\u043C\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u044B\u0445 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0445 \u2014 \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043D\u0438\u0435 \u043E \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u0435 \u0432\u0441\u0435\u0445 \u0432\u043E\u0437\u043C\u043E\u0436\u043D\u044B\u0445 \u0430\u043B\u0433\u0435\u0431\u0440 \u0441 \u0435\u0434\u0438\u043D\u0438\u0446\u0435\u0439, \u0434\u043E\u043F\u0443\u0441\u043A\u0430\u044E\u0449\u0438\u0445 \u043F\u0440\u0438 \u0432\u0432\u0435\u0434\u0435\u043D\u0438\u0438 \u0441\u043A\u0430\u043B\u044F\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0438\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u043F\u0440\u0430\u0432\u0438\u043B\u043E \u00AB\u043D\u043E\u0440\u043C\u0430 \u043F\u0440\u043E\u0438\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u0440\u0430\u0432\u043D\u0430 \u043F\u0440\u043E\u0438\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044E \u043D\u043E\u0440\u043C\u00BB (\u043D\u043E\u0440\u043C\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u0430\u044F \u0430\u043B\u0433\u0435\u0431\u0440\u0430). \u0423\u0441\u0442\u0430\u043D\u043E\u0432\u043B\u0435\u043D\u0430 \u043D\u0435\u043C\u0435\u0446\u043A\u0438\u043C \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u043E\u043C \u0413\u0443\u0440\u0432\u0438\u0446\u0435\u043C \u0432 1898 \u0433\u043E\u0434\u0443.."@ru . . . . . . . . "25167317"^^ . . . . "28196"^^ . . . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0413\u0443\u0440\u0432\u0438\u0446\u0430 \u043E \u043D\u043E\u0440\u043C\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u044B\u0445 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0445 \u0441 \u0434\u0435\u043B\u0435\u043D\u0438\u0435\u043C"@ru . . . . . . . . "\uD6C4\uB974\uBE44\uCE20\uC758 \uC815\uB9AC (\uB098\uB217\uC148 \uB300\uC218)"@ko . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0413\u0443\u0440\u0432\u0456\u0446\u0430 \u043F\u0440\u043E \u043A\u043E\u043C\u043F\u043E\u0437\u0438\u0442\u043D\u0456 \u0430\u043B\u0433\u0435\u0431\u0440\u0438 \u2014 \u0442\u0435\u043E\u0440\u0435\u043C\u0430, \u0449\u043E \u043E\u043F\u0438\u0441\u0443\u0454 \u043E\u0441\u043D\u043E\u0432\u043D\u0456 \u043D\u043E\u0440\u043C\u043E\u0432\u0430\u043D\u0456 \u0430\u043B\u0433\u0435\u0431\u0440\u0438 (\u043D\u0435 \u043F\u043B\u0443\u0442\u0430\u0442\u0438 \u0437 \u043D\u043E\u0440\u043C\u043E\u0432\u0430\u043D\u0438\u043C\u0438 (\u0431\u0430\u043D\u0430\u0445\u043E\u0432\u0438\u043C\u0438) \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u043C\u0438 \u0449\u043E \u0432 \u0444\u0443\u043D\u043A\u0446\u0456\u043E\u043D\u0430\u043B\u044C\u043D\u043E\u043C\u0443 \u0430\u043D\u0430\u043B\u0456\u0437\u0456). \u0426\u044F \u0442\u0435\u043E\u0440\u0435\u043C\u0430 \u0441\u0444\u043E\u0440\u043C\u0443\u043B\u044C\u043E\u0432\u0430\u043D\u0430 \u043D\u0456\u043C\u0435\u0446\u044C\u043A\u0438\u043C \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u043E\u043C \u0410\u0434\u043E\u043B\u044C\u0444\u043E\u043C \u0413\u0443\u0440\u0432\u0456\u0446\u0435\u043C \u0432 1898 \u0440\u043E\u0446\u0456.."@uk . . . . . . "\u80E1\u5C14\u7EF4\u5179\u5B9A\u7406"@zh . . "1092800573"^^ . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0413\u0443\u0440\u0432\u0438\u0446\u0430 \u043E \u043D\u043E\u0440\u043C\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u044B\u0445 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0445 \u2014 \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043D\u0438\u0435 \u043E \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u0435 \u0432\u0441\u0435\u0445 \u0432\u043E\u0437\u043C\u043E\u0436\u043D\u044B\u0445 \u0430\u043B\u0433\u0435\u0431\u0440 \u0441 \u0435\u0434\u0438\u043D\u0438\u0446\u0435\u0439, \u0434\u043E\u043F\u0443\u0441\u043A\u0430\u044E\u0449\u0438\u0445 \u043F\u0440\u0438 \u0432\u0432\u0435\u0434\u0435\u043D\u0438\u0438 \u0441\u043A\u0430\u043B\u044F\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0438\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u043F\u0440\u0430\u0432\u0438\u043B\u043E \u00AB\u043D\u043E\u0440\u043C\u0430 \u043F\u0440\u043E\u0438\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u0440\u0430\u0432\u043D\u0430 \u043F\u0440\u043E\u0438\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044E \u043D\u043E\u0440\u043C\u00BB (\u043D\u043E\u0440\u043C\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u0430\u044F \u0430\u043B\u0433\u0435\u0431\u0440\u0430). \u0423\u0441\u0442\u0430\u043D\u043E\u0432\u043B\u0435\u043D\u0430 \u043D\u0435\u043C\u0435\u0446\u043A\u0438\u043C \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u043E\u043C \u0413\u0443\u0440\u0432\u0438\u0446\u0435\u043C \u0432 1898 \u0433\u043E\u0434\u0443.."@ru . . . . . . . . . . . . . . "Hurwitz's theorem (composition algebras)"@en . . . . . . . . . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0413\u0443\u0440\u0432\u0456\u0446\u0430 \u043F\u0440\u043E \u043A\u043E\u043C\u043F\u043E\u0437\u0438\u0442\u043D\u0456 \u0430\u043B\u0433\u0435\u0431\u0440\u0438"@uk . . . . . . . . "En math\u00E9matiques, diverses versions de th\u00E9or\u00E8mes de Frobenius g\u00E9n\u00E9ralis\u00E9s ont \u00E9tendu progressivement le th\u00E9or\u00E8me de Frobenius de 1877. Ce sont des th\u00E9or\u00E8mes d'alg\u00E8bre g\u00E9n\u00E9rale qui classifient les alg\u00E8bres unif\u00E8res \u00E0 division de dimension finie sur le corps commutatif \u211D des r\u00E9els. Moyennant certaines restrictions, il n'y en a que quatre : \u211D lui-m\u00EAme, \u2102 (complexes), \u210D (quaternions) et \U0001D546 (octonions)."@fr . "Der Quadrate-Satz gibt in der Mathematik an, f\u00FCr welche nat\u00FCrlichen Zahlen das Produkt zweier Summen von quadrierten reellen Zahlen in eine Summe von ebenfalls Quadraten von Zahlen zerf\u00E4llt, die Bilinearformen von ersteren sind. Seit 1818 ist bekannt, dass dies f\u00FCr m\u00F6glich ist und der Kompositionssatz von Adolf Hurwitz aus dem Jahr 1898 besagt, dass dies auch die einzigen sind. Die Normen der reellen und komplexen Zahlen, der Quaternionen und Oktonionen erf\u00FCllen die Relation , woraus sich die bekannten Kompositionen konstruieren lassen. Als direkte Folgerung aus den Identit\u00E4ten ergibt sich, dass die Menge der Summen von Quadratzahlen in den genannten F\u00E4llen bez\u00FCglich der Multiplikation abgeschlossen ist. F\u00FCr war er bereits Diophantos von Alexandria bekannt. Dass er f\u00FCr nicht gilt fand zuerst Adrien-Marie Legendre (in seinem Lehrbuch \u00FCber Zahlentheorie). Den Fall bewies Leonhard Euler 1748 in einem Brief an Goldbach. Der Fall wurde von John T. Graves 1844 im Zusammenhang mit der Theorie der von ihm eingef\u00FChrten Oktaven gefunden (und von Arthur Cayley 1845)."@de . . . . . . . . . . . . "\uD6C4\uB974\uBE44\uCE20\uC758 \uC815\uB9AC(Hurwitz's theorem, -\u5B9A\u7406)\uB294 \uB3C5\uC77C \uC218\uD559\uC790 \uC544\uB3CC\uD504 \uD6C4\uB974\uBE44\uCE20\uC758 \uC774\uB984\uC774 \uBD99\uC740 \uCD94\uC0C1\uB300\uC218\uD559\uC758 \uC815\uB9AC\uB85C, \uD6C4\uB974\uBE44\uCE20\uAC00 1898\uB144 \uC99D\uBA85\uD558\uC600\uB2E4. \uB2E4\uC74C\uACFC \uAC19\uC740 \uB0B4\uC6A9\uC774\uB2E4. \n* \uD56D\uB4F1\uC6D0\uC774 \uC788\uB294 \uC2E4\uC218 \uD639\uC740 \uBCF5\uC18C\uC218 \uC704\uC758 \uB178\uB984\uC774 \uC8FC\uC5B4\uC9C4 \uB85C\uC11C, \uADF8 \uB178\uB984\uC774 \uD56D\uC0C1 \uC744 \uB9CC\uC871\uD558\uB294 \uAC83\uC740 \uD56D\uC0C1 (\uB300\uC218\uB85C\uC11C) \uC2E4\uC218, \uBCF5\uC18C\uC218, \uC0AC\uC6D0\uC218, \uD314\uC6D0\uC218 \uC911 \uD558\uB098\uC640 \uB3D9\uD615\uC774\uB2E4. \uD398\uB974\uB514\uB09C\uD2B8 \uAC8C\uC624\uB974\uD06C \uD504\uB85C\uBCA0\uB2C8\uC6B0\uC2A4\uC5D0 \uC758\uD574 \uC2DC\uC791\uB41C \uC2E4\uC218\uCCB4 \uC704\uC758 \uB098\uB217\uC148 \uB300\uC218\uB97C \uBD84\uB958\uD558\uB294 \uBB38\uC81C\uB294 \uD6C4\uB974\uBE44\uCE20\uAC00 \uC774 \uC815\uB9AC \uB4F1\uC73C\uB85C \uC774\uC5B4\uBC1B\uC544 \uBC1C\uC804\uC2DC\uCF30\uACE0, \uB9C9\uC2A4 \uCD08\uB978\uC774 (alternative ring)\uC758 \uC5F0\uAD6C\uB85C \uC774\uC5D0 \uB300\uD55C \uC77C\uBC18\uC801\uC778 \uACB0\uACFC\uB97C \uC5BB\uC5C8\uB2E4. \uC544\uB3CC\uD504 \uD6C4\uB974\uBE44\uCE20\uB294 \uC774 \uC815\uB9AC\uB97C \uC989\uC2DC \uC751\uC6A9\uD558\uC5EC , \uC624\uC77C\uB7EC\uC758 \uB124 \uC81C\uACF1\uC218 \uD56D\uB4F1\uC2DD\uC774\uB098 \uB370\uAC90\uC758 \uC5EC\uB35F \uC81C\uACF1\uC218 \uD56D\uB4F1\uC2DD\uACFC \uAC19\uC740 \uD56D\uB4F1\uC2DD\uC740 \uBBF8\uC9C0\uC218\uAC00 \uC5EC\uB35F \uAC1C\uBCF4\uB2E4 \uB354 \uB9CE\uC740 \uACBD\uC6B0\uC5D0 \uB300\uD574\uC11C\uB294 \uC131\uB9BD\uD560 \uC218 \uC5C6\uB2E4\uB294 \uC0AC\uC2E4\uC744 \uC99D\uBA85\uD558\uAE30\uB3C4 \uD588\uB2E4."@ko . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0413\u0443\u0440\u0432\u0456\u0446\u0430 \u043F\u0440\u043E \u043A\u043E\u043C\u043F\u043E\u0437\u0438\u0442\u043D\u0456 \u0430\u043B\u0433\u0435\u0431\u0440\u0438 \u2014 \u0442\u0435\u043E\u0440\u0435\u043C\u0430, \u0449\u043E \u043E\u043F\u0438\u0441\u0443\u0454 \u043E\u0441\u043D\u043E\u0432\u043D\u0456 \u043D\u043E\u0440\u043C\u043E\u0432\u0430\u043D\u0456 \u0430\u043B\u0433\u0435\u0431\u0440\u0438 (\u043D\u0435 \u043F\u043B\u0443\u0442\u0430\u0442\u0438 \u0437 \u043D\u043E\u0440\u043C\u043E\u0432\u0430\u043D\u0438\u043C\u0438 (\u0431\u0430\u043D\u0430\u0445\u043E\u0432\u0438\u043C\u0438) \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u043C\u0438 \u0449\u043E \u0432 \u0444\u0443\u043D\u043A\u0446\u0456\u043E\u043D\u0430\u043B\u044C\u043D\u043E\u043C\u0443 \u0430\u043D\u0430\u043B\u0456\u0437\u0456). \u0426\u044F \u0442\u0435\u043E\u0440\u0435\u043C\u0430 \u0441\u0444\u043E\u0440\u043C\u0443\u043B\u044C\u043E\u0432\u0430\u043D\u0430 \u043D\u0456\u043C\u0435\u0446\u044C\u043A\u0438\u043C \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u043E\u043C \u0410\u0434\u043E\u043B\u044C\u0444\u043E\u043C \u0413\u0443\u0440\u0432\u0456\u0446\u0435\u043C \u0432 1898 \u0440\u043E\u0446\u0456.."@uk . "In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859\u20131919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras."@en . . . . . . . "In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859\u20131919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras. The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in . Subsequent proofs of the restrictions on the dimension have been given by using the representation theory of finite groups and by and using Clifford algebras. Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the classical groups and in quantum mechanics to the classification of simple Jordan algebras."@en . . . . . . . . . "Quadrate-Satz"@de . . . . . . . . . . . . . . . "Der Quadrate-Satz gibt in der Mathematik an, f\u00FCr welche nat\u00FCrlichen Zahlen das Produkt zweier Summen von quadrierten reellen Zahlen in eine Summe von ebenfalls Quadraten von Zahlen zerf\u00E4llt, die Bilinearformen von ersteren sind. Seit 1818 ist bekannt, dass dies f\u00FCr m\u00F6glich ist und der Kompositionssatz von Adolf Hurwitz aus dem Jahr 1898 besagt, dass dies auch die einzigen sind. Die Normen der reellen und komplexen Zahlen, der Quaternionen und Oktonionen erf\u00FCllen die Relation , woraus sich die bekannten Kompositionen konstruieren lassen. Als direkte Folgerung aus den Identit\u00E4ten ergibt sich, dass die Menge der Summen von Quadratzahlen in den genannten F\u00E4llen bez\u00FCglich der Multiplikation abgeschlossen ist."@de . "En math\u00E9matiques, diverses versions de th\u00E9or\u00E8mes de Frobenius g\u00E9n\u00E9ralis\u00E9s ont \u00E9tendu progressivement le th\u00E9or\u00E8me de Frobenius de 1877. Ce sont des th\u00E9or\u00E8mes d'alg\u00E8bre g\u00E9n\u00E9rale qui classifient les alg\u00E8bres unif\u00E8res \u00E0 division de dimension finie sur le corps commutatif \u211D des r\u00E9els. Moyennant certaines restrictions, il n'y en a que quatre : \u211D lui-m\u00EAme, \u2102 (complexes), \u210D (quaternions) et \U0001D546 (octonions)."@fr . . . . . . "Th\u00E9or\u00E8me de Frobenius g\u00E9n\u00E9ralis\u00E9"@fr . .