野村, 明人

Kyoto University Research Information Repository

Title分岐条件の付いた埋め込み問題とその応用について(代数的整数論における最近の話題)Author(s)野村, 明人Citation数理解析研究所講究録 (1992), 797: 17-24Issue Date1992-08URL http://hdl.handle.net/2433/82784RightType Departmental Bulletin PaperTextversionpublisherKyoto University17(Akito Nomura), $p$. , [31 [51. , $p$ .\S 1 Introduction$k$ , $\otimes$ $k$ , $k$ $k$ ,. $K/k$ , $(\epsilon)$ : $1arrow Z/pZarrow Earrow Ga1(K/k)arrow 1$. $(K/k, \epsilon)$ ,$\otimes$$\downarrow\varphi$$(*)$$j$$(\epsilon):1arrow Z/pZarrow Earrow Ga1(K/k)$ –\rightarrow $1$. $\varphi$ . $\otimes$ $E$ $\psi$ ,$jo\psi=\varphi$ $(K/k, \epsilon)$ solution , solution$(K/k, e)$ . , solution $\psi$ , $\psi$ propersolution . , solution $\psi$ , $Ker\psi$ $k$$(K/k, e)$ solution field . $(K/k, e)$ proper solution ,797 1992 17-2418$L/K/k$ $1arrow Ga1(L/K)arrow Ga1(L/k)arrow Ga1(K/k)arrow 1$$(\epsilon)$ .$k$$q$ , $q$ $k$ $k_{q},$ $k$ $K$ $K$$K_{q}$ . , $(K/k, e)$ $(K_{q}/k_{q}, e_{q})$.$\oplus_{q}$$\downarrow\varphi|c_{q}$$(\epsilon_{\sigma})$ : 1 $arrow Z/pZarrow E_{q}arrow^{j1_{B_{\eta}}}G_{q}arrow 1$ .$G_{q}$ $K_{q}/k_{q}$ , $6_{q}$ $k_{q}$ , $E_{q}$ $G_{q}$ $j$. , $(K_{q}/k_{q}, e_{q})$ $s$olution, solution field , $(K/k, e)$., .Fact 1(Ikeda[l]) $(K/k, e)$ . $(K/k, e)$proper solution .Fact 2(Neukirch[2]) $k$ $P$ $(K_{\mathfrak{p}}/k_{\mathfrak{p}}, e_{P})$ .$(K/k, e)$ .Fact 3(Neukirch[2]) $(K/k, e)$ . $S$ $k$, $S$ $q$ $L(q)$ $(K_{q}/k_{q}, e_{q})$ solution field .19$(K/k, e)$ solution field $L$ , $S$ $q$ $L$ $L(q)$.\S 2 Main theorem$(K/k, e)$ , $(*)$ .$(K/k, e, \phi)$ .Definition $(K/k, e, \emptyset)$ $(K/k, e)$ $(*)$ . $\emptyset$ $E$$\psi$ $(K/k, e, \emptyset)$ solution ,(1) $\psi$ $(K/k, e)$ solution,(2) $\psi$ solution field $K$ ,. $\psi$ , $\psi$ proper solution ..Theorem 1 $K/k/Q$(1) $[k : Q]$ $p$ ,(2) $K/k$ $p$ ,, $(e)$ : $1arrow Z/pZarrow Earrow Ga1(K/Q)arrow 1$ , split. $(K/Q, \epsilon, \emptyset)$ proper solution .( ) , , $(K/Q, e)$ .(1)(2) , Fact 3 $(K/Q, e)$ solution field $L_{1}$ $L_{1}/K$ ,20$p$ . $L_{1}$ , $(K/Q, e)$ propersolution . , $L_{1}$ soluiton $\psi$ , $\emptyset,$ $Ga1(K/Q),$ $E$ $p-$Sylow $\emptyset(p),$ $G(p),$ $E(p)$ , $H^{2}(Ga1(K/Q), Z/pZ)arrow$$H^{2}(G(p), Z/pZ)$ 1 1 , $1arrow Z/pZarrow E(p)arrow G(p)arrow 1$split . , $E(p)$ $G(p)$, $\psi|_{\emptyset(p)}$ : $\emptyset(p)arrow E(p)$ . $\psi$ : $\emptysetarrow E$ , $\psi$$(K/Q, e)$ proper solution . $L_{1}/K$ $L_{1}$. $L_{1}$ $q_{1}$ $L_{1}/K$ . $q_{1}$ $K,$ $Q$$q,$ $q$ , $K_{q}$ $Q_{q}$ $P$ $F$$1arrow Ga1(L_{1q_{1}}/K_{q})arrow Ga1(L_{1q_{1}}/F)arrow Ga1(K_{q}/F)arrow 1$ split , $q$ $F$$\overline{q}$ $F$ $p$ . $N_{F/Q_{q}}(\overline{q})\equiv 1(mod. p)$ . $F/Q_{q}$$P$ $q\equiv 1(mod. p)$ .$T$ $Q(\zeta_{q})$ $p$ , $\sim_{q}$ $q$ $T\cdot L_{1}$ , $\overline{q}$ $T\cdot L_{1}/K$$L_{2}$ . $L_{2}$$(a)L_{2}$ $(K/Q, e)$ solution field,$(b)L_{2}/K$ $q$ ,$(c)L_{1}/K$ $L_{2},/K$ ,. . ( )21\S 3 Applications, Theorem 1 . 2$p$ .Theorem 2 $k$ 2 , $p$ , $k$ -rank 2. $L/k$ $Ga1(L/k)$$\{\alpha, \gamma|\alpha^{p}=\beta^{p}=\gamma^{p}=1, \gamma^{-1}\alpha\gamma=\alpha\beta,\beta^{-1}\alpha\beta=\alpha,\beta^{-1}\gamma\beta=\gamma\}$.( ) , $K/k$ $(Z/pZ)^{2}$. $K$ $Q$ $Ga1(K/Q)$$G=\{\tau\iota,$ $v,$ $w|u^{p}=v^{p}=w^{2}=1,$ $w^{-1}uw=u^{-1},$ $w^{-1}vw=v^{-1},$ $v^{-1}uv=u\rangle$.$E=\{\begin{array}{llllll} \alpha^{p} =\beta^{p}=\gamma^{p}=\delta =1,\gamma \alpha\gamma=\alpha\beta,\beta \alpha\beta=\alpha\alpha,\gamma_{)}\delta| -1\end{array}\}$$\beta^{-1}\gamma\beta=\gamma,$ $\delta^{-1}\alpha\delta=\alpha^{-1},$ $\delta^{-1}\beta 5=\beta,$ $\delta^{-1}\gamma\delta=\gamma$, split$j$$1arrow<\beta>arrow Earrow Garrow 1$. $j$ $\alpha$ }$arrow u,$ $\gamma-\rangle$ $v,$ $\delta-w$ .Theorem 1 $(K/Q, e, \phi)$ ( proper solution $\psi$ . $\psi$solution field $L$ $L/k$ . ( )22Hilbert $p$ $p$ .Theorem 3 $l,$ $p$ , $p^{f}\equiv 1(mod. p)$ $f$. , $/Q$ $l$ . $k$$p$ , $k$ $P$ $P$.( ) $k$ $p$ , $K/k/Q$(a)(b) .(a) $K/k$ $p$ .(b) $K’/k/Q$ $\subsetneq K’\subsetneq K$ .Gal( /Q) Gal(K/ ) , $F_{p}$ [Gal( /Q)], -rank Gal(K/ ) $=fl^{j}$ $i$(cf.[5;Theorem 9]) ,$H^{2}( Ga1(K/k), Z/pZ)\cong(Z/pZ)\frac{fl^{j}(fl^{j}+1)}{2}$ .$f$$p \frac{fl^{j}(fl^{j}+1)}{2}\not\equiv 1$ (mod.l).,$H^{2}(Ga1(K/Q), Z/pZ)\cong H^{2}(Ga1(K/k), Z/pZ)^{Ga1(k/Q)}$ .23Gal( /Q) $H^{2}$ (Gal(K/ ), $Z/pZ$ )$H^{2}(Ga1(K/$ $), Z/pZ)^{Ga1(k/Q)}\neq 0$,$H^{2}(Ga1(K/Q), Z/pZ)\neq 0$. split$(e):1arrow Z/pZarrow Earrow Ga1(K/Q)arrow 1$, Theorem 1 $L_{1}/K/Q$ $Ga1(L_{1}/Q)$ $E$$L_{1}/k$ $p$ . $L_{1}/k$$H^{2}(Ga1(L_{1}/Q), Z/pZ)\neq 0$. $L/k/Q$, $L/k$ $p$ .$p$ $p$ . ( )Remark. $p$ , $p$ $Q$ 3(cf. [6] [7]). Theorem3 .24References$[1].M.Ikeda$ , Zur Existenz eigentlicher galoisscher Korper beim Einbettungsprob-lem, Hamb.Abh.$24(1960),126-131$ .[2].J.Neukirch, Uber das Einbettungsproblem der algebraischen Zahlentheorie,Invent. Math.$21(1973),59-116$ .$[3].A.Nomura$ , On the existence of unramified p-extensions, Osaka J.Math.28(1991), 55-62$[4].A.Nomura$, On imbedding problems with ramification conditions and theirapplications, Dissertation, Kanazawa University,1991$[5].A.Nomura$, A remark on the class number of Hilbert’s class fields (preprint).$[6].K.Uchida$ , Class numbers of cubic cyclic fields, J.Math.Soc.Japan 26 No.3(1974), 447-453.[7].L.Wasington, Class Numbers of the Simplest Cubic Fields, Math.Comp.48(1987), 371-384.

ブンキ ジョウケン ノ ツイタ ウメコミ モンダイ ト ソノ オウヨウ ニ ツイテ ダイスウテキ セイスウロン ニ オケル サイキン ノ ワダイ

http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/82784/1/0797-02.pdf

ブンキジョウケンノツイタウメコミモンダイトソノオウヨウニツイテダイスウテキセイスウロンニオケルサイキンノワダイ

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository

ブンキ ジョウケン ノ ツイタ ウメコミ モンダイ ト ソノ オウヨウ ニ ツイテ ダイスウテキ セイスウロン ニ オケル サイキン ノ ワダイ

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository

ブンキジョウケンノツイタウメコミモンダイトソノオウヨウニツイテダイスウテキセイスウロンニオケルサイキンノワダイ