{"id":718,"date":"2020-04-13T10:57:36","date_gmt":"2020-04-13T02:57:36","guid":{"rendered":"http:\/\/kylelv.com\/?p=718"},"modified":"2020-04-13T10:57:36","modified_gmt":"2020-04-13T02:57:36","slug":"%ce%bb-%e7%9f%a9%e9%98%b5%e6%80%a7%e8%b4%a8%e6%8e%a8%e5%b9%bf%e5%88%b0%e6%95%b4%e6%95%b0%e7%8e%af%e4%b8%8a%e7%9f%a9%e9%98%b5","status":"publish","type":"post","link":"https:\/\/blog.kylelv.com\/?p=718","title":{"rendered":"\u03bb-\u77e9\u9635\u6027\u8d28\u63a8\u5e7f\u5230\u6574\u6570\u73af\u4e0a\u77e9\u9635"},"content":{"rendered":"\n
\u6574\u6570\u73af\u4e0a\u7684\u77e9\u9635\u6027\u8d28<\/a>\u4e0b\u8f7d<\/a><\/div>\n\n\n\n

\uff08\u7b2c\u4e00\u6b21\u4f7f\u7528LaTeX\uff0c\u987a\u4fbf\u7eaa\u5ff5\u4e00\u4e0bqaq<\/p>\n\n\n\n

\\documentclass[UTF8]{ctexart}\n\n\\pagestyle{plain} \n\\ctexset{\n    section = {\n        name=\\S\n    }\n}\n\\usepackage{amsmath,amssymb,amsthm,mathrsfs}\n\\usepackage{enumerate}\n\\newtheorem{theorem}{{\u5b9a\u7406}} \n\\newtheorem{proposition}{{\u547d\u9898}} \n\\newtheorem*{lemma*}{{\u5f15\u7406}}\n\\newtheorem{lemma}{{\u5f15\u7406}}\n\\newtheorem{corollary}{{\u63a8\u8bba}}[theorem] \n\\newtheorem{definition}{{\u5b9a\u4e49}} \n\n\n\\title{\u6574\u6570\u73af\u4e0a\u7684\u77e9\u9635\u6027\u8d28}\n\\author{1900013053 \u5415\u6606\u822a\\ \u4fe1\u606f\u79d1\u5b66\u6280\u672f\u5b66\u9662}\n\\date{\\today}\n\n\\begin{document}\n\\maketitle\n\\tableofcontents\n\\newpage\n\\section{\u6574\u6570\u77e9\u9635}\n    \\begin{definition}\n        \u5982\u679c\u6574\u6570\u77e9\u9635$A$ \u4e2d\u6709\u4e00\u4e2a$r(r \\geq 1)$\u7ea7\u5b50\u5f0f\u4e0d\u4e3a\u96f6\uff0c\u800c\u6240\u6709$r+1$\u7ea7\u5b50\u5f0f\n        \uff08\u5982\u679c\u6709\u7684\u8bdd\uff09\u5168\u4e3a\u96f6\uff0c\u5219\u79f0$A$\u7684\u79e9\u4e3a$r$.\u96f6\u77e9\u9635\u7684\u79e9\u89c4\u5b9a\u4e3a\u96f6.\n    \\end{definition}\n    \\begin{definition}\n        \u4e00\u4e2a$n*n$\u7684\u6574\u6570\u77e9\u9635$A$ \u79f0\u4e3a\u5728\u6574\u6570\u73af\u53ef\u9006\u7684\uff08\u4e0b\u7b80\u8bb0\u4e3a\u53ef\u9006\u7684\uff09\uff0c\u5982\u679c\u6709\u4e00\u4e2a$n*n$\u7684\u6574\u6570\u77e9\u9635$B$\u4f7f\n        \\begin{equation}\n            AB=BA=E\uff0c\n        \\end{equation}\n        \u8fd9\u91cc$E$\u662f$n$\u7ea7\u5355\u4f4d\u77e9\u9635.\u9002\u5408$(1)$\u7684\u77e9\u9635$B$\uff08\u5b83\u662f\u552f\u4e00\u7684\uff09\u79f0\u4e3a$A$\u7684\u9006\u77e9\u9635\uff0c\u8bb0\u4e3a$A^{-1}$.\n    \\end{definition}\n    \\begin{theorem}\n        \u4e00\u4e2a$n*n$\u7684\u6574\u6570\u77e9\u9635$A$\u662f\u53ef\u9006\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u4e3a\u884c\u5217\u5f0f$|A|$\u662f$\\pm 1$.\n    \\end{theorem}\n    \\begin{proof}\n        \u6709$A$\u5728\u6574\u6570\u73af\u4e0a\u53ef\u9006\uff0c\u5219\u5b58\u5728\u6574\u6570\u77e9\u9635$B$\u6ee1\u8db3\\begin{equation*}\n            AB=BA=E,|AB|=|E|=1\n        \\end{equation*}\n        \u800c\\begin{equation*}\n            |AB|=|A|\\cdot |B|=1,|A|\\in Z,|B|\\in Z\n        \\end{equation*}\n        \u56e0\u6b64$|A|=\\pm 1$.\n        \\newline\n        \u82e5$|A|=\\pm 1$\uff0c\u5219$A$\u5728\u5b9e\u6570\u57df$R$\u4e0a\u53ef\u9006\uff0c\u4e14\\begin{equation*}\n            A^{-1}=\\pm A^*\n        \\end{equation*}\n        \u7531\u4e8e$A$\u4e3a\u6574\u6570\u77e9\u9635\uff0c\u6240\u4ee5$A^*$\u4e3a\u6574\u6570\u77e9\u9635\uff0c\u4ece\u800c$A^{-1}$\u4e3a\u6574\u6570\u77e9\u9635\uff0c\u5373$A$\u5728\u6574\u6570\u73af\u53ef\u9006.\n    \\end{proof}\n\n\\section{\u6574\u6570\u77e9\u9635\u5728\u521d\u7b49\u53d8\u6362\u4e0b\u7684\u6807\u51c6\u5f62}\n    \\begin{definition}\n        \u4e0b\u9762\u4e09\u79cd\u53d8\u6362\u53eb\u505a\u6574\u6570\u77e9\u9635\u7684\u521d\u7b49\u53d8\u6362\uff1a\n        \\begin{enumerate}[1)]\n            \\item \u77e9\u9635\u7684\u4e24\u884c\uff08\u5217\uff09\u4e92\u6362\u4f4d\u7f6e\uff1b\n            \\item \u77e9\u9635\u7684\u67d0\u4e00\u884c\uff08\u5217\uff09\u4e58\u5e38\u6570$-1$\uff1b\n            \\item \u77e9\u9635\u7684\u67d0\u4e00\u884c\uff08\u5217\uff09\u52a0\u53e6\u4e00\u884c\uff08\u5217\uff09\u7684k\u500d\uff0ck\u4e3a\u6574\u6570.\n        \\end{enumerate}\n    \\end{definition}\n    \\begin{definition}\n        \u6574\u6570\u77e9\u9635$A$\u79f0\u4e3a\u4e0e$B$\u7b49\u4ef7\uff0c\u5982\u679c\u53ef\u4ee5\u901a\u8fc7\u4e00\u7cfb\u5217\u521d\u7b49\u53d8\u6362\u5c06$A$\u5316\u4e3a$B$.\n    \\end{definition}\n    \\begin{lemma*}\n        \u8bbe\u6574\u6570\u77e9\u9635$A$\u7684\u5de6\u4e0a\u89d2\u5143\u7d20$a_{11}\\neq 0$\uff0c\u5e76\u4e14A\u4e2d\u81f3\u5c11\u6709\u4e00\u4e2a\u5143\u7d20\u4e0d\u80fd\u88ab\u5b83\u9664\u5c3d\uff0c\n        \u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u627e\u5230\u4e00\u4e2a\u4e0e$A$\u7b49\u4ef7\u7684\u77e9\u9635$B$\uff0c\u4ed6\u7684\u5de6\u4e0a\u89d2\u5143\u7d20\u4e5f\u4e0d\u4e3a\u96f6\uff0c\u4f46\u662f\u7edd\u5bf9\u503c\u5c0f\u4e8e$a_{11}$.\n    \\end{lemma*}\n    \\begin{proof}\n        \u6839\u636e$A$\u4e2d\u4e0d\u80fd\u88ab$a_{11}$\u9664\u5c3d\u7684\u5143\u7d20\u6240\u5728\u4f4d\u7f6e\uff0c\u5206\u4e09\u79cd\u60c5\u51b5\u8ba8\u8bba\uff1a\n        \\begin{enumerate}[1)]\n            \\item \u82e5\u5728$A$\u7684\u7b2c\u4e00\u5217\u4e2d\u6709\u4e00\u4e2a\u5143\u7d20$a_{11}$\u9664\u5c3d\uff0c\u5219\u6709\\begin{equation*}\n                    a_{i1}=a_{11}q+r,\n                \\end{equation*}\n                \u5176\u4e2d\u4f59\u6570$r\\neq 0$\uff0c\u4e14\u5c0f\u4e8e$a_{11}$.\n                \\newline\n                \u5bf9$A$\u4f5c\u521d\u7b49\u53d8\u6362.\u628a$A$\u7684\u7b2c$i$\u884c\u51cf\u53bb\u7b2c\u4e00\u884c\u7684$q$\u500d\uff0c\u5f97\\begin{equation*}\n                    A=\\begin{pmatrix}\n                        a_{11}& &\\cdots \\\\\n                        \\vdots& &\\vdots \\\\\n                        a_{i1}& &\\cdots \\\\\n                        \\vdots& &\\vdots \n                    \\end{pmatrix}\n                    \\to \\begin{pmatrix}\n                        a_{11}& &\\cdots \\\\\n                        \\vdots& &\\vdots \\\\\n                        r& &\\cdots \\\\\n                        \\vdots& &\\vdots \n                    \\end{pmatrix},\n                \\end{equation*}\n                \u518d\u5c06\u6b64\u77e9\u9635\u7684\u7b2c\u4e00\u884c\u4e0e\u7b2ci\u884c\u4e92\u6362\u5f97\uff0c\\begin{equation*}\n                    A\\to \\begin{pmatrix}\n                        r& &\\cdots \\\\\n                        \\vdots& &\\vdots \\\\\n                        a_{11}& &\\cdots \\\\\n                        \\vdots& &\\vdots \n                    \\end{pmatrix}=B .\n                \\end{equation*}\n                $B$\u7684\u5de6\u4e0a\u89d2\u5143\u7d20$r$\u7b26\u5408\u5f15\u7406\u7684\u8981\u6c42\uff0c\u6545$B$\u5373\u4e3a\u6240\u6c42\u77e9\u9635.\n            \\item \u5728$A$\u7684\u7b2c\u4e00\u884c\u4e2d\u6709\u4e00\u4e2a\u5143\u7d20$a_{1i}$\u4e0d\u80fd\u88ab$a_{11}$\u9664\u5c3d\uff0c\u8fd9\u79cd\u60c5\u51b5\u7684\u8bc1\u660e\u4e0e$1)$\u7c7b\u4f3c\uff0c\u4f46\u662f\u5bf9$A$\u8fdb\u884c\u7684\u662f\u521d\u7b49\u5217\u53d8\u6362.\n            \\item $A$\u7684\u7b2c\u4e00\u884c\u4e0e\u7b2c\u4e00\u5217\u4e2d\u7684\u5143\u7d20\u90fd\u53ef\u88ab$a_{11}$\u9664\u5c3d\uff0c\u4f46$A$\u4e2d\u6709\u53e6\u4e00\u4e2a\u5143\u7d20$a_{ij}(i>1,j>1)$\n                \u4e0d\u80fd\u88ab$a_{11}$\u9664\u5c3d.\u6211\u4eec\u8bbe$a_{i1}=k a_{11}$.\u5bf9$A$\u4f5c\u4e0b\u8ff0\u521d\u7b49\u53d8\u6362\uff1a\\begin{equation*}\\begin{aligned}\n                    A=\\begin{pmatrix}\n                        a_{11}&\\cdots&a_{1j}&\\cdots \\\\\n                        \\vdots& &\\vdots&\\vdots \\\\\n                        a_{i1}&\\cdots&a_{ij}&\\cdots \\\\\n                        \\vdots& &\\vdots&\\vdots\n                    \\end{pmatrix} \\to \\begin{pmatrix}\n                        a_{11}&\\cdots&a_{1j}&\\cdots \\\\\n                        \\vdots& &\\vdots&\\vdots \\\\\n                        0&\\cdots&a_{ij}-ka_{1j}&\\cdots \\\\\n                        \\vdots& &\\vdots&\\vdots\n                    \\end{pmatrix} \\\\\n                    \\to \\begin{pmatrix}\n                        a_{11}&\\cdots&a_{ij}+(1-k)a_{1j}&\\cdots \\\\\n                        \\vdots& &\\vdots&\\vdots \\\\\n                        0&\\cdots&a_{ij}-ka_{1j}&\\cdots \\\\\n                        \\vdots& &\\vdots&\\vdots\n                    \\end{pmatrix} =A_1 .\n                \\end{aligned}\\end{equation*}\n                \u77e9\u9635$A_1$\u7684\u7b2c\u4e00\u884c\u4e2d\uff0c\u6709\u4e00\u4e2a\u5143\u7d20\\begin{equation*}\n                    a_{ij}+(1-k)a_{1j}\n                \\end{equation*}\n                \u4e0d\u80fd\u88ab\u5de6\u4e0a\u89d2\u5143\u7d20$a_{11}$\u9664\u5c3d\uff0c\u8fd9\u5c31\u5316\u4e3a\u5df2\u7ecf\u8bc1\u660e\u4e86\u7684\u60c5\u51b5$2)$.\n        \\end{enumerate}\n    \\end{proof}\n    \\begin{theorem}\n        \u4efb\u610f\u4e00\u4e2a\u975e\u96f6\u7684$s*n$\u7684\u6574\u6570\u77e9\u9635$A$\u90fd\u7b49\u4ef7\u4e8e\u4e0b\u5217\u5f62\u5f0f\u7684\u77e9\u9635\\begin{equation*}\n            \\begin{pmatrix}\n                d_1& & & & & & \\\\\n                 &d_2& & & & & \\\\\n                 & &\\ddots& & & & \\\\\n                 & & &d_r& & & \\\\\n                 & & & &0& & \\\\\n                 & & & & &\\ddots& \\\\\n                 & & & & & &0\n            \\end{pmatrix},\n        \\end{equation*}\n        \u5176\u4e2d$r\\geq 1,d_i(i=1,2,\\cdots,r)$\u4e3a\u6b63\u6574\u6570\uff0c\u4e14\\begin{equation*}\n            d_i\\mid d_{i+1} \\ (i=1,2,\\cdots,r-1) .\n        \\end{equation*}\n    \\end{theorem}\n    \\begin{proof}\n        \u7ecf\u8fc7\u884c\u5217\u8c03\u52a8\u4e4b\u540e\uff0c\u53ef\u4ee5\u4f7f\u5f97$A$\u7684\u5de6\u4e0a\u89d2\u5143\u7d20$a_{11}\\neq 0$\uff0c\u5982\u679c$a_{11}$\u4e0d\u80fd\u9664\u5c3d$A$\u7684\u5168\u90e8\u5143\u7d20\uff0c\n        \u7531\u5f15\u7406\uff0c\u53ef\u4ee5\u627e\u5230\u4e0e$A$\u7b49\u4ef7\u7684$B_1$\uff0c\u5b83\u7684\u5de6\u4e0a\u89d2\u5143\u7d20$b_1 \\neq 0$\uff0c\u5e76\u4e14\u5c0f\u4e8e$a_{11}$.\u5982\u679c$b_1$\u8fd8\u4e0d\u80fd\u9664\u5c3d$B_1$\u7684\u5168\u90e8\u5143\u7d20\uff0c\n        \u7531\u5f15\u7406\uff0c\u53c8\u53ef\u4ee5\u627e\u5230\u4e0e$B_1$\u7b49\u4ef7\u7684$B_2$\uff0c\u5b83\u7684\u5de6\u4e0a\u89d2\u5143\u7d20$b_2 \\neq 0$\uff0c\u5e76\u4e14\u5c0f\u4e8e$b_1$.\n        \u5982\u6b64\u4e0b\u53bb\uff0c\u5c06\u5f97\u5230\u4e00\u7cfb\u5217\u5f7c\u6b64\u7b49\u4ef7\u7684\u6574\u6570\u77e9\u9635\uff0c$A,B_1,B_2,\\cdots$.\u5b83\u4eec\u7684\u5de6\u4e0a\u89d2\u5143\u7d20\u90fd\u4e0d\u4e3a\u96f6\uff0c\u800c\u4e14\u4f9d\u6b21\u51cf\u5c0f.\u4f46\u7531\u4e8e\u5176\u4e3a\u6b63\u6574\u6570\uff0c\u4e0d\u80fd\u65e0\u6b62\u5883\u51cf\u5c0f\uff0c\u4e14$1$\u80fd\u4e58\u9664\u6240\u6709\u6570.\n        \u56e0\u6b64\u5728\u6709\u9650\u6b65\u4ee5\u540e\uff0c\u6211\u4eec\u5c06\u7ec8\u6b62\u4e8e\u4e00\u4e2a\u6574\u6570\u77e9\u9635$B_s$\uff0c\u5b83\u7684\u5de6\u4e0a\u89d2\u5143\u7d20$b_s \\neq 0$\uff0c\u800c\u4e14\u53ef\u4ee5\u9664\u5c3d$B_s$\u7684\u5168\u90e8\u5143\u7d20$b_{ij}$\uff0c\u5373\\begin{equation*}\n            b_{ij}=b_s q_{ij},\n        \\end{equation*}\n        \u5bf9$B_s$\u4f5c\u521d\u7b49\u53d8\u6362\uff1a\\begin{equation*}\n            B_s=\\begin{pmatrix}\n                b_s&\\cdots&b_{ij}&\\cdots\\\\\n                \\vdots& &\\vdots&\\vdots\\\\\n                b_{i1}&\\cdots&\\cdots&\\cdots\\\\\n                \\vdots& &\\vdots&\\vdots\n            \\end{pmatrix} \\to \\begin{pmatrix}\n                b_s&0&\\cdots&0\\\\\n                0& & & \\\\\n                \\vdots& &A_1& \\\\\n                0& & & \n            \\end{pmatrix}\n        \\end{equation*}\n        \u5728\u53f3\u4e0b\u89d2\u7684\u6574\u6570\u77e9\u9635$A_1$\u4e2d\uff0c\u5168\u90e8\u5143\u7d20\u90fd\u662f\u53ef\u4ee5\u88ab$b_s$\u9664\u5c3d\u7684\uff0c\u56e0\u4e3a\u5b83\u4eec\u90fd\u662f$B_s$\u4e2d\u5143\u7d20\u7684\u7ec4\u5408.\n        \\newline\n        \u5982\u679c$A_1 \\neq O$\uff0c\u5219\u5bf9\u4e8e$A_1$\u53ef\u4ee5\u91cd\u590d\u4e0a\u8ff0\u8fc7\u7a0b\uff0c\u8fdb\u800c\u628a\u77e9\u9635\u5316\u6210\\begin{equation*}\n            \\begin{pmatrix}\n                d_1&0&\\ \\cdots &0\\\\\n                0&d_2&\\ \\cdots &0\\\\\n                0&0& & \\\\\n                \\vdots&\\vdots&\\ A_2& \\\\\n                0&0& & \n            \\end{pmatrix},\n        \\end{equation*}\n        \u5176\u4e2d$d_1$\u4e0e$d_2$\u90fd\u662f\u6b63\u6574\u6570\uff0c\u800c\u4e14$d_1 \\mid d_2$\uff0c$d_2$\u80fd\u9664\u5c3d$A_2$\u7684\u5168\u90e8\u5143\u7d20.\n        \\newline\n        \u5982\u6b64\u4e0b\u53bb\uff0c$A$\u6700\u540e\u5c31\u5316\u6210\u4e86\u6240\u8981\u6c42\u7684\u5f62\u5f0f.\n    \\end{proof}\n    \u6700\u540e\u5316\u6210\u7684\u8fd9\u4e2a\u77e9\u9635\u79f0\u4e3a$A$\u7684\u6807\u51c6\u5f62.\n\\section{\u4e0d\u53d8\u56e0\u5b50}\n    \\begin{definition}\n        \u8bbe\u6574\u6570\u77e9\u9635$A$\u7684\u79e9\u4e3a$r$\uff0c\u5bf9\u4e8e\u6b63\u6574\u6570$k,1 \\leq k \\leq r$\uff0c$A$\u4e2d\u5fc5\u6709\u975e\u96f6\u7684$k$\u7ea7\u5b50\u5f0f.\n        $A$\u4e2d\u5168\u90e8$k$\u7ea7\u5b50\u5f0f\u7684\u6700\u5927\u516c\u56e0\u6570$D_k$\u79f0\u4e3a$A$\u7684$k$\u7ea7\u884c\u5217\u5f0f\u56e0\u5b50.\n    \\end{definition}\n    \\begin{theorem}\n        \u7b49\u4ef7\u7684\u6574\u6570\u77e9\u9635\u5177\u6709\u76f8\u540c\u7684\u79e9\u4e0e\u76f8\u540c\u7684\u5404\u7ea7\u884c\u5217\u5f0f\u56e0\u5b50.\n    \\end{theorem}\n    \\begin{proof}\n        \u6211\u4eec\u53ea\u9700\u8981\u8bc1\u660e\uff0c\u6574\u6570\u77e9\u9635\u7ecf\u8fc7\u4e00\u6b21\u521d\u7b49\u53d8\u6362\uff0c\u79e9\u4e0e\u884c\u5217\u5f0f\u56e0\u5b50\u662f\u4e0d\u53d8\u7684.\n        \\newline\n        \u8bbe\u6574\u6570\u77e9\u9635$A$\u7ecf\u8fc7\u4e00\u6b21\u521d\u7b49\u884c\u53d8\u6362\u53d8\u6210$B$\uff0c$p,q$\u5206\u522b\u4e3a$A$\u4e0e$B$\u7684$k$\u7ea7\u884c\u5217\u5f0f\u56e0\u5b50.\u6211\u4eec\u8bc1\u660e$p=q$.\u4e0b\u9762\u5206\u4e09\u79cd\u60c5\u51b5\u8ba8\u8bba\uff1a\n        \\begin{enumerate}[1)]\n            \\item $A$\u7ecf\u8fc7\u7b2c\u4e00\u79cd\u521d\u7b49\u884c\u53d8\u6362\u53d8\u6210$B$.\u8fd9\u65f6\uff0c$B$\u7684\u6bcf\u4e00\u4e2a$k$\u7ea7\u5b50\u5f0f\u6216\u8005\u7b49\u4e8e$A$\u7684\u67d0\u4e2a$k$\u7ea7\u5b50\u5f0f\uff0c\u6216\u8005\u4e0e$A$\u7684\u67d0\u4e00\u4e2a$k$\u7ea7\u5b50\u5f0f\u53cd\u53f7\uff0c\n                \u56e0\u6b64$p$\u662f$B$\u7684$k$\u7ea7\u5b50\u5f0f\u7684\u516c\u56e0\u5b50\uff0c\u4ece\u800c$p \\mid q$.\n            \\item $A$\u7ecf\u8fc7\u7b2c\u4e8c\u79cd\u521d\u7b49\u884c\u53d8\u6362\u53d8\u6210$B$.\u8fd9\u65f6\uff0c$B$\u7684\u6bcf\u4e00\u4e2a$k$\u7ea7\u5b50\u5f0f\u6216\u8005\u7b49\u4e8e$A$\u7684\u67d0\u4e2a$k$\u7ea7\u5b50\u5f0f\uff0c\u6216\u8005\u4e0e$A$\u7684\u67d0\u4e00\u4e2a$k$\u7ea7\u5b50\u5f0f\u53cd\u53f7\uff0c\u540c\u7406\u53ef\u5f97$p \\mid q$.\n            \\item $A$\u7ecf\u8fc7\u7b2c\u4e09\u79cd\u521d\u7b49\u884c\u53d8\u6362\u53d8\u6210$B$.\u8fd9\u65f6$B$\u4e2d\u90a3\u4e9b\u5305\u542b$i$\u884c\u4e0e$j$\u884c\u7684$k$\u7ea7\u5b50\u5f0f\u548c\u90a3\u4e9b\u4e0d\u5305\u542b$i$\u884c\u7684$k$\u7ea7\u5b50\u5f0f\u90fd\u7b49\u4e8e$A$\u4e2d\u5bf9\u5e94\u7684$k$\u7ea7\u5b50\u5f0f\uff1b\n                $B$\u4e2d\u90a3\u4e9b\u5305\u542b$i$\u884c\u4f46\u4e0d\u5305\u542b$j$\u884c\u7684$k$\u7ea7\u5b50\u5f0f\uff0c\u6309$i$\u884c\u5206\u6210\u4e24\u90e8\u5206\uff0c\u800c\u7b49\u4e8e$A$\u7684\u4e00\u4e2a$k$\u7ea7\u5b50\u5f0f\u4e0e\u53e6\u4e00\u4e2a$k$\u7ea7\u5b50\u5f0f\u7684$\\pm c$\u500d\u7684\u548c\uff0c\n                \u4e5f\u5c31\u662f$A$\u7684\u4e24\u4e2a$k$\u7ea7\u5b50\u5f0f\u7684\u7ec4\u5408.\u56e0\u6b64$p$\u662f$B$\u7684$k$\u7ea7\u5b50\u5f0f\u7684\u516c\u56e0\u5b50\uff0c\u4ece\u800c$p \\mid q$.\n        \\end{enumerate}\n        \u5bf9\u4e8e\u5217\u53d8\u6362\uff0c\u53ef\u4ee5\u5b8c\u5168\u4e00\u6837\u5730\u8ba8\u8bba.\u603b\u4e4b\uff0c\u5982\u679c$A$\u7ecf\u8fc7\u4e00\u6b21\u521d\u7b49\u53d8\u6362\u53d8\u6210$B$\uff0c\u90a3\u4e48$p \\mid q$.\n        \u4f46\u7531\u4e8e\u521d\u7b49\u53d8\u6362\u7684\u53ef\u9006\u6027\uff0c$B$\u4e5f\u53ef\u4ee5\u7ecf\u8fc7\u4e00\u6b21\u521d\u7b49\u53d8\u6362\u53d8\u6210$A$.\u7531\u4e0a\u9762\u7684\u8ba8\u8bba\uff0c\u540c\u6837\u5e94\u6709$q \\mid p$\uff0c\u4e8e\u662f$p=q$.\n        \\newline\n        \u5f53$A$\u7684\u5168\u90e8$k$\u7ea7\u5b50\u5f0f\u4e3a\u96f6\u65f6\uff0c$B$\u7684\u5168\u90e8$k$\u7ea7\u5b50\u5f0f\u4e5f\u5c31\u7b49\u4e8e\u96f6\uff1b\u53cd\u4e4b\u4ea6\u7136.\u56e0\u6b64\uff0c$A$\u4e0e$B$\u65e2\u6709\u76f8\u540c\u7684\u5404\u7ea7\u884c\u5217\u5f0f\u56e0\u5b50\uff0c\u53c8\u6709\u76f8\u540c\u7684\u79e9.\n    \\end{proof}\n    \\begin{theorem}\n        \u6574\u6570\u77e9\u9635\u7684\u6807\u51c6\u5f62\u662f\u552f\u4e00\u7684.\n    \\end{theorem}\n    \\begin{proof}\n        \\setcounter{equation}{0}\n        \u8bbe\\begin{equation}\n            \\begin{pmatrix}\n                d_1& & & & & & \\\\\n                 &d_2& & & & & \\\\\n                 & &\\ddots& & & & \\\\\n                 & & &d_r& & & \\\\\n                 & & & &0& & \\\\\n                 & & & & &\\ddots& \\\\\n                 & & & & & &0\n            \\end{pmatrix}\n        \\end{equation}\n        \u662f$A$\u7684\u6807\u51c6\u5f62.\u7531\u4e8e$A$\u4e0e$(1)$\u7b49\u4ef7\uff0c\u5b83\u4eec\u6709\u76f8\u540c\u7684\u79e9\u4e0e\u76f8\u540c\u7684\u884c\u5217\u5f0f\u56e0\u5b50\uff0c\u56e0\u6b64\uff0c\n        $A$\u7684\u79e9\u5c31\u662f\u6807\u51c6\u5f62\u7684\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u975e\u96f6\u5143\u7d20\u7684\u4e2a\u6570$r$\uff1b$A$\u7684k\u7ea7\u884c\u5217\u5f0f\u56e0\u5b50\u5c31\u662f\\begin{equation}\n            D_k=d_1 d_2 \\cdots d_k \\ (k=1,2,\\cdots,r).\n        \\end{equation}\n        \u4e8e\u662f\\begin{equation}\n            d_1=D_1,d_2=\\frac{D_2}{D_1},\\cdots,d_r=\\frac{D_r}{D_{r-1}}.\n        \\end{equation}\n        \u8fd9\u8bf4\u660e$A$\u7684\u6807\u51c6\u5f62$(1)$\u7684\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u7684\u975e\u96f6\u5143\u7d20\u662f\u88ab$A$\u7684\u884c\u5217\u5f0f\u56e0\u5b50\u6240\u552f\u4e00\u786e\u5b9a\u7684\uff0c\u6240\u4ee5$A$\u7684\u6807\u51c6\u5f62\u662f\u552f\u4e00\u7684.\n    \\end{proof}\n    \\begin{definition}\n        \u6807\u51c6\u5f62\u7684\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u7684\u975e\u96f6\u5143\u7d20$d_1,d_2,\\cdots,d_r$\u79f0\u4e3a\u6574\u6570\u77e9\u9635$A$\u7684\u4e0d\u53d8\u56e0\u5b50.\n    \\end{definition}\n    \\begin{theorem}\n        \u4e24\u4e2a\u6574\u6570\u77e9\u9635\u7b49\u4ef7\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u5b83\u4eec\u6709\u76f8\u540c\u7684\u884c\u5217\u5f0f\u56e0\u5b50\uff0c\u6216\u8005\uff0c\u5b83\u4eec\u6709\u76f8\u540c\u7684\u4e0d\u53d8\u56e0\u5b50.\n    \\end{theorem}\n    \\begin{proof}\n        \u7b49\u5f0f$(2)$\u4e0e$(3)$\u7ed9\u51fa\u4e86\u6574\u6570\u77e9\u9635\u7684\u884c\u5217\u5f0f\u56e0\u5b50\u4e0e\u4e0d\u53d8\u56e0\u5b50\u4e4b\u95f4\u7684\u5173\u7cfb.\u8fd9\u4e2a\u5173\u7cfb\u8bf4\u660e\uff0c\u884c\u5217\u5f0f\u56e0\u5b50\u4e0e\u4e0d\u53d8\u56e0\u5b50\u662f\u76f8\u4e92\u786e\u5b9a\u7684.\n        \u56e0\u6b64\uff0c\u8bf4\u4e24\u4e2a\u77e9\u9635\u6709\u76f8\u540c\u7684\u5404\u7ea7\u884c\u5217\u5f0f\u56e0\u5b50\uff0c\u5c31\u7b49\u4e8e\u8bf4\u5b83\u4eec\u6709\u76f8\u540c\u7684\u5404\u7ea7\u4e0d\u53d8\u56e0\u5b50.\n        \\newline \u5fc5\u8981\u6027\u5df2\u6709\u5b9a\u74063 \u8bc1\u660e.\n        \\newline \u5145\u5206\u6027\u662f\u5f88\u663e\u7136\u7684.\n        \u4e8b\u5b9e\u4e0a\uff0c\u82e5\u6574\u6570\u77e9\u9635$A$\u4e0e$B$\u6709\u76f8\u540c\u7684\u4e0d\u53d8\u56e0\u5b50\uff0c\u5219$A$\u4e0e$B$\u548c\u540c\u4e00\u6807\u51c6\u5f62\u7b49\u4ef7\uff0c\u56e0\u800c$A$\u4e0e$B$\u7b49\u4ef7.\n    \\end{proof}\n    \\begin{theorem}\n        \u77e9\u9635$A$\u662f\u53ef\u9006\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u5b83\u53ef\u4ee5\u8868\u793a\u6210\u4e00\u4e9b\u521d\u7b49\u77e9\u9635\u7684\u4e58\u79ef.\n    \\end{theorem}\n    \\begin{proof}\n        \u8bbe$A$\u4e3a\u4e00\u4e2a$n*n$\u53ef\u9006\u77e9\u9635\uff0c\u7531\u5b9a\u74061\u77e5$|A|=\\pm 1$\uff0c\u8fd9\u5c31\u662f\u8bf4\uff0c$D_n=1$.\u4ece\u800c\u53ef\u77e5\uff0c$d_k=1 \\ (k=1,2,\\cdots,n)$.\n        \u56e0\u6b64\u53ef\u9006\u77e9\u9635\u7684\u6807\u51c6\u5f62\u4e3a\u5355\u4f4d\u77e9\u9635$E$.\u53cd\u8fc7\u6765\uff0c\u4e0e\u5355\u4f4d\u77e9\u9635\u7b49\u4ef7\u7684\u77e9\u9635\u4e00\u5b9a\u662f\u53ef\u9006\u7684\uff0c\u56e0\u4e3a\u4ed6\u7684\u884c\u5217\u5f0f\u662f\u4e00\u4e2a\u975e\u96f6\u7684\u6570.\n        \u8fd9\u5c31\u662f\u8bf4\uff0c\u77e9\u9635\u53ef\u9006\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u5b83\u4e0e\u5355\u4f4d\u77e9\u9635\u7b49\u4ef7.\u53c8\u77e9\u9635$A$\u4e0e$B$\u7b49\u4ef7\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u6709\u4e00\u7cfb\u5217\u521d\u7b49\u77e9\u9635$P_1,P_2,\\cdots,P_l,Q_1,Q_2,\\cdots,Q_t$\uff0c\u4f7f\\begin{equation*}\n            A=P_1P_2\\cdots P_l B Q_1Q_2\\cdots Q_t\n        \\end{equation*}.\n        \u7279\u522b\u5730\uff0c\u5f53$B=E$\u65f6\uff0c\u5b9a\u7406\u6210\u7acb.\n    \\end{proof}\n    \\begin{corollary}\n        \u4e24\u4e2a$s*n$\u7684\u6574\u6570\u77e9\u9635$A$\u4e0e$B$\u7b49\u4ef7\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u4e3a\uff0c\u6709\u4e00\u4e2a$s*s$\u53ef\u9006\u77e9\u9635$P$\u4e0e\u4e00\u4e2a$n*n$\u53ef\u9006\u77e9\u9635$Q$\uff0c\u4f7f\\begin{equation*}\n            B=PAQ .\n        \\end{equation*}\n    \\end{corollary}\n\\section{\u521d\u7b49\u56e0\u5b50}\n    \\begin{definition}\n        \u628a\u77e9\u9635$A$\uff08\u6216\u7ebf\u6027\u53d8\u6362$\\mathscr{A}$\uff09\u7684\u6bcf\u4e2a\u6b21\u6570\u5927\u4e8e\u96f6\u7684\u4e0d\u53d8\u56e0\u5b50\u5206\u89e3\u6210\u4e92\u7d20\u7684\u8d28\u56e0\u5b50\u65b9\u5e42\u7684\u4e58\u79ef\uff0c\n        \u6240\u6709\u8fd9\u4e9b\u8d28\u56e0\u5b50\u65b9\u5e42\u79f0\u4e3a\u77e9\u9635$A$\u7684\u521d\u7b49\u56e0\u5b50.\n    \\end{definition}\n    \\begin{theorem}\n        \u4e24\u4e2a\u7684\u6574\u6570\u77e9\u9635\u7b49\u4ef7\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u5b83\u4eec\u6709\u76f8\u540c\u79e9\u548c\u76f8\u540c\u7684\u521d\u7b49\u56e0\u5b50.\n    \\end{theorem}\n    \\begin{proof}\n        \u82e5\u4e24\u4e2a\u6574\u6570\u77e9\u9635\u7b49\u4ef7\uff0c\u5219\u5b83\u4eec\u5b58\u5728\u76f8\u540c\u7684\u4e0d\u53d8\u56e0\u5b50\uff0c\u56e0\u6b64\uff0c\u5b83\u4eec\u6709\u76f8\u540c\u79e9\u548c\u76f8\u540c\u7684\u521d\u7b49\u56e0\u5b50.\n        \\newline\n        \u5047\u8bbe\u77e9\u9635A\u7684\u4e0d\u53d8\u56e0\u5b50\u5df2\u77e5\uff0c\u5c06$d_i\\ (i=1,2,\\cdots,n)$\u5206\u89e3\u6210\u8d28\u56e0\u5b50\u65b9\u5e42\u7684\u4e58\u79ef\uff1a\\begin{equation*}\\begin{aligned}\n            d_1=p_1^{k_{11}}p_2^{k_{12}}\\cdots p_r^{k_{1r}},\\\\\n            d_2=p_1^{k_{21}}p_2^{k_{22}}\\cdots p_r^{k_{2r}},\\\\\n            \\cdots\\cdots\\ \\ \\ \\ \\ \\ \\ \\ \\ \\\\\n            d_n=p_1^{k_{n1}}p_2^{k_{n2}}\\cdots p_r^{k_{nr}},\n        \\end{aligned}\\end{equation*}\n        \u6211\u4eec\u6ce8\u610f\u5230\\begin{equation*}\n            d_i \\mid d_{i+1},\n        \\end{equation*}\u4ece\u800c\\begin{equation*}\n            p_j^{k_{ij}} \\mid p_j^{k_{i+1,j}} \\ (i=1,2,\\cdots,n-1;j=1,2,\\cdots,r).\n        \\end{equation*}\n        \u56e0\u6b64\u5728$d_i\\ (i=1,2,\\cdots,n)$\u5206\u89e3\u5f0f\u4e2d\uff0c\u5c5e\u4e8e\u540c\u4e00\u4e2a\u8d28\u56e0\u5b50\u7684\u65b9\u5e42\u7684\u6307\u6570\u6709\u9012\u589e\u7684\u6027\u8d28.\u8fd9\u8bf4\u660e\uff0c\u540c\u4e00\u4e2a\u8d28\u56e0\u5b50\u65b9\u5e42\u4f5c\u6210\u7684\u521d\u7b49\u56e0\u5b50\u4e2d\uff0c\n        \u65b9\u5e42\u6700\u9ad8\u7684\u5fc5\u5b9a\u51fa\u73b0\u5728$d_n$\u7684\u5206\u89e3\u4e2d.\u5982\u6b64\u987a\u63a8\u4e0b\u53bb\uff0c\u53ef\u77e5\u5c5e\u4e8e\u540c\u4e00\u4e2a\u8d28\u56e0\u5b50\u65b9\u5e42\u7684\u521d\u7b49\u56e0\u5b50\u5728\u4e0d\u53d8\u56e0\u5b50\u7684\u5206\u89e3\u5f0f\u4e2d\u51fa\u73b0\u7684\u4f4d\u7f6e\u662f\u552f\u4e00\u786e\u5b9a\u7684.\n        \\newline\n        \u8bbe\u4e00\u4e2a\u77e9\u9635\u7684\u5168\u90e8\u521d\u7b49\u56e0\u5b50\u4e3a\u5df2\u77e5\uff0c\u5728\u5168\u90e8\u521d\u7b49\u56e0\u5b50\u4e2d\u5c06\u540c\u4e00\u4e2a\u8d28\u56e0\u5b50\u7684\u65b9\u5e42\u7684\u90a3\u4e9b\u521d\u7b49\u56e0\u5b50\u6309\u964d\u5e42\u6392\u5217\uff0c\u4e2a\u6570\u4e0d\u8db3$n$\u65f6\uff0c\u5c31\u5728\u540e\u9762\u8865\u4e0a\u9002\u5f53\u4e2a\u6570\u7684$1$\uff0c\u4f7f\u5f97\u51d1\u6210$n$\u4e2a.\n        \u8bbe\u6240\u5f97\u6392\u5217\u4e3a\\begin{equation*}\n            p_j^{k_{nj}},p_j^{k_{n-1,j}},\\cdots,p_j^{k_{1j}} \\ (j=1,2,\\cdots,r).\n        \\end{equation*}\u4e8e\u662f\u4ee4\\begin{equation*}\n            d_i=p_1^{k_{i1}}p_2^{k_{i2}}\\cdots p_r^{k_{ir}} \\ (i=1,2,\\cdots,n),\n        \\end{equation*}\n        \u5219$d_i\\ (i=1,2,\\cdots,n)$\u5c31\u662fA\u7684\u4e0d\u53d8\u56e0\u5b50.\n        \\newline\n        \u56e0\u6b64\uff0c\u82e5\u4e24\u4e2a\u6574\u6570\u77e9\u9635\u6709\u76f8\u540c\u7684\u79e9\u548c\u521d\u7b49\u56e0\u5b50\uff0c\u5219\u5b83\u4eec\u5c31\u6709\u76f8\u540c\u7684\u4e0d\u53d8\u56e0\u5b50\uff0c\u56e0\u6b64\u4ed6\u4eec\u7b49\u4ef7.\n    \\end{proof}\n    \\begin{lemma*}\n        \u8bbe\\begin{equation*}\n            A=\\begin{pmatrix}\n                a_1 b_1&0\\\\\n                0&a_2 b_2\n            \\end{pmatrix},\n            B=\\begin{pmatrix}\n                a_2 b_1&0\\\\\n                0&a_1 b_2\n            \\end{pmatrix}.\n        \\end{equation*}\n        \u5982\u679c\u6b63\u6574\u6570$a_1,a_2$\u90fd\u4e0e$b_1,b_2$\u4e92\u7d20\uff0c\u5219$A$\u4e0e$B$\u7b49\u4ef7.\n    \\end{lemma*}\n    \\begin{proof}\n        \u663e\u7136$A$\u548c$B$\u6709\u76f8\u540c\u7684\u4e8c\u7ea7\u884c\u5217\u5f0f\u56e0\u5b50.\u800c$A$\u548c$B$\u7684\u4e00\u7ea7\u884c\u5217\u5f0f\u56e0\u5b50\u5206\u522b\u4e3a\\begin{equation*}\n            d=(a_1b_1,a_2,b_2)\n        \\end{equation*}\u548c\\begin{equation*}\n            d'=(a_2b_1,a_1b_2).\n        \\end{equation*}\n        \u6613\u77e5\\begin{equation*}\n            d=(a_1,a_2)(b_1,b_2)=d',\n        \\end{equation*}\n        \u56e0\u800c$A$\u548c$B$\u4e5f\u6709\u76f8\u540c\u7684\u4e00\u7ea7\u884c\u5217\u5f0f\u56e0\u5b50.\u6240\u4ee5$A$\u4e0e$B$\u7b49\u4ef7.\n    \\end{proof}\n    \\begin{theorem}\n        \u9996\u5148\u7528\u521d\u7b49\u53d8\u6362\u5316\u6574\u6570\u77e9\u9635$A$\u4e3a\u5bf9\u89d2\u5f62\u5f0f\uff0c\u7136\u540e\u5c06\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u5206\u89e3\u6210\u4e92\u4e0d\u76f8\u540c\u7684\u8d28\u56e0\u5b50\u65b9\u5e42\u7684\u4e58\u79ef\uff0c\n        \u5219\u6240\u6709\u8fd9\u4e9b\u8d28\u56e0\u5b50\u7684\u65b9\u5e42\uff08\u76f8\u540c\u7684\u6309\u51fa\u73b0\u7684\u6b21\u6570\u8ba1\u7b97\uff09\u5c31\u662f$A$\u7684\u5168\u90e8\u521d\u7b49\u56e0\u5b50.\n    \\end{theorem}\n    \\begin{proof}\n        \u8bbe\u6574\u6570\u77e9\u9635$A$\u5df2\u7528\u521d\u7b49\u53d8\u6362\u5316\u4e3a\u5bf9\u89d2\u5f62\u5f0f\\begin{equation*}\n            D=\\begin{pmatrix}\n               a_1& & & \\\\\n                &a_2& & \\\\\n                & &\\cdots& \\\\\n                & & &a_n \n            \\end{pmatrix},\n        \\end{equation*}\n        \u5176\u4e2d$a_i$\u5747\u4e3a\u6b63\u6574\u6570.\u5c06$a_i$\u5206\u89e3\u4e3a\u4e92\u4e0d\u76f8\u540c\u7684\u8d28\u56e0\u5b50\u65b9\u5e42\u7684\u4e58\u79ef\uff1a\\begin{equation*}\n            a_i=p_1^{k_{i1}}p_2^{k_{i2}}\\cdots p_r^{k_{ir}} \\ (i=1,2,\\cdots,n).\n        \\end{equation*}\n        \u6211\u4eec\u73b0\u5728\u8981\u8bc1\u660e\u7684\u662f\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u76f8\u540c\u7684\u8d28\u56e0\u5b50\u7684\u65b9\u5e42$p_j^{k_{1j}},p_j^{k_{2j}},\\cdots,p_j^{k_{nj}}(j=1,2,\\cdots,r)$\n        \u5728$D$\u7684\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u6309\u9012\u5347\u5e42\u6b21\u6392\u5217\u540e\uff0c\u5f97\u5230\u7684\u65b0\u5bf9\u89d2\u77e9\u9635$D'$\u4e0e$D$\u7b49\u4ef7.\u6b64\u65f6$D'$\u5c31\u662f$A$\u7684\u6807\u51c6\u5f62\u800c\u4e14\u6240\u6709\u4e0d\u4e3a$1$\u7684$p_j^{k_{ij}}$\u5c31\u662f$A$\u7684\u5168\u90e8\u521d\u7b49\u56e0\u5b50.\n        \\newline\n        \u4e3a\u65b9\u4fbf\u8d77\u89c1\uff0c\u5148\u5bf9$p_1$\u7684\u65b9\u5e42\u8fdb\u884c\u8ba8\u8bba.\u4ee4\\begin{equation*}\n            b_i=p_2^{k_{i2}}p_3^{k_{i3}}\\cdots p_r^{k_{ir}},\\ i=1,2,\\cdots,n,\n        \\end{equation*}\u4e8e\u662f\\begin{equation*}\n            a_i=p_1^{k_{i1}}b_i,\\ i=1,2,\\cdots,n,\n        \\end{equation*}\n        \u800c\u4e14\u6bcf\u4e2a$p_1^{k_{i1}}$\u90fd\u4e0e$b_i$\u4e92\u7d20.\u5982\u679c\u6709\u76f8\u90bb\u7684\u4e00\u5bf9\u6307\u6570$k_{i1}>k_{i+1,1}$\uff0c\n        \u5219\u5728$D$\u4e2d\u5c06$p_1^{k_{i1}}$\u4e0e$p_1^{k_{i+1,1}}$\u5bf9\u8c03\u4f4d\u7f6e\uff0c\u800c\u5176\u4ed6\u56e0\u5f0f\u4fdd\u6301\u4e0d\u52a8.\u6839\u636e\u5f15\u7406\uff0c\\begin{equation*}\n            \\begin{pmatrix}\n                p_1^{k_{i1}}b_i&0\\\\\n                0&p_1^{k_{i+1,1}}b_{i+1}\n            \\end{pmatrix}\n        \\end{equation*}\u4e0e\\begin{equation*}\n            \\begin{pmatrix}\n                p_1^{k_{i+1,1}}b_i&0\\\\\n                0&p_1^{k_{i1}}b_{i+1}\n            \\end{pmatrix}\n        \\end{equation*}\u7b49\u4ef7.\n        \u4ece\u800c$D$\u4e0e\u5bf9\u89d2\u77e9\u9635\\begin{equation*}\n            D_1=\\begin{pmatrix}\n                p_1^{k_{11}}b_1& & & & & \\\\\n                 &\\ddots& & & & \\\\\n                 & &p_1^{k_{i+1,1}}b_i& & & \\\\\n                 & & &p_1^{k_{i1}}b_{i+1}& & \\\\\n                 & & & &\\ddots& \\\\\n                 & & & & &p_1^{k_{n1}}b_n\n            \\end{pmatrix}\n        \\end{equation*}\u7b49\u4ef7.\n        \u7136\u540e\u5bf9$D_1$\u4f5c\u5982\u4e0a\u8ba8\u8bba.\u5982\u6b64\u7ee7\u7eed\u8fdb\u884c\uff0c\u76f4\u5230\u5bf9\u89d2\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u5143\u7d20\u6240\u542b$p_1$\u7684\u65b9\u5e42\u662f\u6309\u9012\u5347\u5e42\u6b21\u6392\u5217\u4e3a\u6b62.\n        \u4f9d\u6b21\u5bf9$p_2,\\cdots,p_r$\u4f5c\u540c\u6837\u5904\u7406\uff0c\u6700\u540e\u4fbf\u5f97\u5230\u4e0e$D$\u7b49\u4ef7\u7684\u5bf9\u89d2\u77e9\u9635$D'$\uff0c\n        \u5b83\u7684\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u6240\u542b\u6bcf\u4e2a\u76f8\u540c\u7684\u8d28\u56e0\u5b50\u7684\u65b9\u5e42\uff0c\u90fd\u662f\u6309\u9012\u5347\u5e42\u6b21\u6392\u5217\u7684.\n    \\end{proof}\n\\end{document} \n<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
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