翻訳と辞書 Words near each other ・ Ramian County ・ Ramichloridium musae ・ Ramicourt ・ RAMiCS ・ Ramidanda ・ Ramidava ・ Ramidreju ・ Ramie ・ Ramie Dowling ・ Ramie Leahy ・ Ramiel ・ Ramiele Malubay ・ Ramification ・ Ramification (botany) ・ Ramification (mathematics) ・ Ramification group ・ Ramification problem ・ Ramification theory of valuations ・ Ramifications (Ligeti) ・ Ramified forcing ・ Ramiflory ・ Ramihrdus of Cambrai ・ Ramik Wilson ・ Ramil Aritkulov ・ Ramil Gallego ・ Ramil Ganiyev ・ Ramil Garifullin ・ Ramil Gasimov ・ Ramil Guliyev ・ Ramil Kharisov Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Ramification group ： ウィキペディア英語版 Ramification group In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension. == Ramification groups in lower numbering == Ramification groups are a refinement of the Galois group $G$ of a finite $L/K$ Galois extension of local fields. We shall write $w,\; \backslash mathcal\; O\_L,\; \backslash mathfrak\; p$ for the valuation, the ring of integers and its maximal ideal for $L$. As a consequence of Hensel's lemma, one can write $\backslash mathcal\; O\_L\; =\; \backslash mathcal\; O\_K()$ for some $\backslash alpha\; \backslash in\; L$ where $O\_K$ is the ring of integers of $K$.〔Neukirch (1999) p.178〕 (This is stronger than the primitive element theorem.) Then, for each integer $i\; \backslash ge\; -1$, we define $G\_i$ to be the set of all $s\; \backslash in\; G$ that satisfies the following equivalent conditions. *(i) $s$ operates trivially on $\backslash mathcal\; O\_L\; /\; \backslash mathfrak\; p^.$ *(ii) $w(s(x)\; -\; x)\; \backslash ge\; i+1$ for all $x\; \backslash in\; \backslash mathcal\; O\_L$ *(iii) $w(s(\backslash alpha)\; -\; \backslash alpha)\; \backslash ge\; i+1.$ The group $G\_i$ is called ''$i$-th ramification group''. They form a decreasing filtration, :$G\_\; =\; G\; \backslash supset\; G\_0\; \backslash supset\; G\_1\; \backslash supset\; \backslash dots\; \backslash .$ In fact, the $G\_i$ are normal by (i) and trivial for sufficiently large $i$ by (iii). For the lowest indices, it is customary to call $G\_0$ the inertia subgroup of $G$ because of its relation to splitting of prime ideals, while $G\_1$ the wild inertia subgroup of $G$. The quotient $G\_0\; /\; G\_1$ is called the tame quotient. The Galois group $G$ and its subgroups $G\_i$ are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular, *$G/G\_0\; =\; \backslash operatorname(l/k),$ where $l,\; k$ are the (finite) residue fields of $L,\; K$.〔since $G/G\_0$ is canonically isomorphic to the decomposition group.〕 *$G\_0\; =\; 1\; \backslash Leftrightarrow\; L/K$ is unramified. *$G\_1\; =\; 1\; \backslash Leftrightarrow\; L/K$ is tamely ramified (i.e., the ramification index is prime to the residue characteristic.) The study of ramification groups reduces to the totally ramified case since one has $G\_i\; =\; (G\_0)\_i$ for $i\; \backslash ge\; 0$. One also defines the function $i\_G(s)\; =\; w(s(\backslash alpha)\; -\; \backslash alpha),\; s\; \backslash in\; G$. (ii) in the above shows $i\_G$ is independent of choice of $\backslash alpha$ and, moreover, the study of the filtration $G\_i$ is essentially equivalent to that of $i\_G$.〔Serre (1979) p.62〕 $i\_G$ satisfies the following: for $s,\; t\; \backslash in\; G$, *$i\_G(s)\; \backslash ge\; i\; +\; 1\; \backslash Leftrightarrow\; s\; \backslash in\; G\_i.$ *$i\_G(t\; s\; t^)\; =\; i\_G(s).$ *$i\_G(st)\; \backslash ge\; \backslash min\backslash .$ Fix a uniformizer $\backslash pi$ of $L$. Then $s\; \backslash mapsto\; s(\backslash pi)/\backslash pi$ induces the injection $G\_i/G\_\; \backslash to\; U\_/U\_,\; i\; \backslash ge\; 0$ where $U\_\; =\; \backslash mathcal\_L^\backslash times,\; U\_\; =\; 1\; +\; \backslash mathfrak^i$. (The map actually does not depend on the choice of the uniformizer.〔Conrad〕) It follows from this〔Use $U\_/U\_\; \backslash simeq\; l^\backslash times$ and $U\_/U\_\; \backslash approx\; l^+$〕 *$G\_0/G\_1$ is cyclic of order prime to $p$ *$G\_i/G\_$ is a product of cyclic groups of order $p$. In particular, $G\_1$ is a ''p''-group and $G$ is solvable. The ramification groups can be used to compute the different $\backslash mathfrak\_$ of the extension $L/K$ and that of subextensions:〔Serre (1979) 4.1 Prop.4, p.64〕 :$w(\backslash mathfrak\_)\; =\; \backslash sum\_\; i\_G(s)\; =\; \backslash sum\_0^\backslash infty\; (|G\_i|\; -\; 1).$ If $H$ is a normal subgroup of $G$, then, for $\backslash sigma\; \backslash in\; G$, $i\_(\backslash sigma)\; =\; i\_G(s)$.〔Serre (1979) 4.1. Prop.3, p.63〕 Combining this with the above one obtains: for a subextension $F/K$ corresponding to $H$, :$v\_F(\backslash mathfrak\_)\; =\; i\_G(s).$ If $s\; \backslash in\; G\_i,\; t\; \backslash in\; G\_j,\; i,\; j\; \backslash ge\; 1$, then $sts^t^\; \backslash in\; G\_$.〔Serre (1979) 4.2. Proposition 10.〕 In the terminology of Lazard, this can be understood to mean the Lie algebra $\backslash operatorname(G\_1)\; =\; \backslash sum\_\; G\_i/G\_$ is abelian. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ramification group」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.