\u8bbe$m,n\\neq1$\u662f\u4e92\u4e0d\u76f8\u540c\u7684\u65e0\u5e73\u65b9\u56e0\u5b50\u7684\u6574\u6570\u3002\u53cc\u4e8c\u6b21\u57df$K=\\Q[\\sqrt m,\\sqrt n]$\u662f$\\Q$\u7684\u6b63\u89c4\u6269\u5f20\uff0c\u5176\u4f3d\u7f57\u74e6\u7fa4$G\\cong V_4$\u3002\u5176\u4e09\u4e2a\u4e8c\u6b21\u5b50\u57df\u6211\u4eec\u8bb0\u4e3a$F_1,F_2,F_3$\uff0c\u5bf9\u5e94\u7684\u4f3d\u7f57\u74e6\u7fa4\u8bb0\u4e3a$H_1,H_2,H_3$\u3002\u8bbe$p$\u662f$\\Z$\u4e0a\u7684\u4e00\u4e2a\u7d20\u6570\uff0c$\\mathfrak{P}$\u662f$\\mathcal{O}_K$\u4e2d$p$\u4e4b\u4e0a\u7684\u7d20\u7406\u60f3\u3002\u8bbe$G_\\mathfrak{P}\\subseteq G$\u4e3a$\\mathfrak{P}$\u7684\u5206\u89e3\u7fa4\uff0c$I_\\mathfrak{P}\\subseteq G_\\mathfrak{P}$\u4e3a$\\mathfrak{P}$\u7684\u60ef\u6027\u7fa4\u3002\u5982\u4e0b\u5173\u7cfb\u6210\u7acb\uff1a
\\[
\\Q\\subseteq K^{G_\\mathfrak{P}}\\subseteq K^{I_\\mathfrak{P}}\\subseteq K
\\]
<\/p>\n\n\n\n
\u5df2\u77e5\u60ef\u6027\u57df$K^{I_\\mathfrak{P}}$\u662f$K$\u4e2d\u6ee1\u8db3$p$\u4e0d\u5206\u6b67\u7684\u6700\u5927\u5b50\u57df\uff0c\u53c8$p$\u5728$F_1,F_2,F_3$\u4e2d\u90fd\u5206\u6b67\uff0c\u6211\u4eec\u6709$F_1\\nsubseteq K^{I_\\mathfrak{P}},F_2\\nsubseteq K^{I_\\mathfrak{P}},F_3\\nsubseteq K^{I_\\mathfrak{P}}$\u3002<\/p>\n\n\n\n
\u8003\u8651\u514b\u83b1\u56e0\u56db\u5143\u7fa4\u7684\u4f3d\u7f57\u74e6\u5bf9\u5e94\uff0c$K$\u7684\u5b50\u57df\u53ea\u6709$\\Q,F_1,F_2,F_3,K$\u3002\u5047\u8bbe$K^{I_\\mathfrak{P}}=K$\uff0c\u5219$p$\u5728\u6574\u4e2a$K$\u4e2d\u90fd\u4e0d\u5206\u6b67\uff0c\u77db\u76fe\u3002\u6545$K^{I_\\mathfrak{P}}=\\Q$\u3002\u6211\u4eec\u5f97\u5230\u60ef\u6027\u7fa4$I_{\\mathfrak{P}}=\\mathrm{Gal}(K\/\\Q)=G$\uff0c\u6240\u4ee5\u5206\u6b67\u6307\u6570$e=\\left| I_{\\mathfrak{P}}\\right|=4$\uff0c\u5373$p$\u5728$K$\u4e0a\u5b8c\u5168\u5206\u6b67\u3002<\/p>\n\n\n\n
\u5df2\u77e5\u5206\u89e3\u57df$K^{G_\\mathfrak{P}}$\u662f$K$\u4e2d\u6ee1\u8db3$p$\u5b8c\u5168\u5206\u88c2\u7684\u6700\u5927\u5b50\u57df\uff0c\u53c8$p$\u5728$F_1,F_2,F_3$\u4e2d\u90fd\u5b8c\u5168\u5206\u88c2\uff0c\u6211\u4eec\u6709$F_1\\subseteq K^{G_\\mathfrak{P}},F_2\\subseteq K^{G_\\mathfrak{P}},F_3\\subseteq K^{G_\\mathfrak{P}}$\uff0c\u5e76\u5f97\u5230$F_1\\cdot F_2\\cdot F_3\\subseteq K^{G_\\mathfrak{P}}$\u3002<\/p>\n\n\n\n
\u8003\u8651\u514b\u83b1\u56e0\u56db\u5143\u7fa4\u7684\u4f3d\u7f57\u74e6\u5bf9\u5e94\uff0c\u6211\u4eec\u6709$F_1\\cdot F_2\\cdot F_3=K^{G_\\mathfrak{P}}$\uff0c\u4e8e\u662f$K^{G_\\mathfrak{P}}=K$\u3002\u6211\u4eec\u5f97\u5230\u5206\u89e3\u7fa4$G_{\\mathfrak{P}}=\\mathrm{Gal}(K\/K)=\\left\\{1\\right\\}$\uff0c\u6240\u4ee5$e\\cdot f=\\left|G_{\\mathfrak{P}}\\right|=1$\uff0c\u5373$p$\u5728$K$\u4e0a\u5b8c\u5168\u5206\u88c2\u3002<\/p>\n\n\n\n
\u5047\u8bbe$p$\u5728$F_i$\u4e0a\u4fdd\u6301\u60ef\u6027\uff0c\u5176\u4e2d$i\\in\\{1,2,3\\}$\uff0c\u5219$p$\u5728$F_i$\u4e0a\u4e0d\u5206\u6b67\uff0c\u56e0\u6b64$F_i\\subseteq K^{I_{\\mathfrak{P}}}$\u3002\u540c\u4e0a\uff0c\u6211\u4eec\u5f97\u5230$K^{I_{\\mathfrak{P}}}=K$\uff0c\u4e8e\u662f$I_{\\mathfrak{P}}=\\mathrm{Gal}(K\/K)=\\left\\{1\\right\\}$\u3002<\/p>\n\n\n\n
\u53c8$p$\u5728$F_i$\u4e0a\u4fdd\u6301\u60ef\u6027\uff0c$p$\u5728$F_i$\u4e0a\u4e0d\u5206\u88c2\uff0c\u610f\u5473\u7740$F_i\\nsubseteq K^{G_\\mathfrak{P}}$\u3002\u540c\u4e0a\uff0c$K$\u7684\u5b50\u57df\u53ea\u6709$\\Q,F_1,F_2,F_3,K$\uff0c\u5047\u8bbe$K^{G_\\mathfrak{P}}=K$\uff0c\u5219$p$\u5728\u6574\u4e2a$K$\u4e2d\u90fd\u5b8c\u5168\u5206\u88c2\uff0c\u77db\u76fe\u3002\u6545$K^{G_\\mathfrak{P}}=\\Q$\u3002\u6211\u4eec\u5f97\u5230\u5206\u89e3\u7fa4$G_{\\mathfrak{P}}=\\mathrm{Gal}(K\/\\Q)=G$\uff0c\u4e8e\u662f\u6709\uff1a
\\[
\\mathrm{Gal}\\left((\\mathcal{O}_K\/\\mathfrak{P})\/\\mathbb{F}_p\\right)=G_\\mathfrak{P}\/I_{\\mathfrak{P}}=G\\cong V_4
\\]
\u77db\u76fe\uff0c\u56e0\u4e3a\u6709\u9650\u57df\u7684\u6709\u9650\u6269\u5f20\u7684\u4f3d\u7f57\u74e6\u7fa4\u4e00\u5b9a\u662f\u5faa\u73af\u7684\uff0c\u800c$V_4$\u975e\u5faa\u73af\u3002<\/p>\n\n\n\n
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