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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Capitulation (algebra) ： ウィキペディア英語版 Principalization (algebra) In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when given an extension of algebraic number fields, some ideal (or more generally fractional ideal) of the ring of integers of the smaller field isn't principal but it's extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field (which can always be generated by at most two elements) become principal when extended to the larger field. In 1897 David Hilbert conjectured that maximal abelian unramified extension of the base field, which was later called the Hilbert class field of the given base field, is such an extension. This conjecture, now known as principal ideal theorem, was proved by Philipp Furtwängler in 1930 after it had been translated from number theory to group theory by Emil Artin in 1929, who made use of his general reciprocity law to establish the reformulation. Since this long desired proof was achieved by means of Artin transfers of non-abelian groups with derived length two, several investigators tried to exploit the theory of such groups further to obtain additional information on the principalization in intermediate fields between the base field and its Hilbert class field. The first contributions in this direction are due to Arnold Scholz and Olga Taussky in 1934, who coined the synonym capitulation for principalization. Another independent access to the principalization problem via Galois cohomology of unit groups is also due to Hilbert and goes back to the chapter on cyclic extensions of number fields of prime degree in his number report, which culminates in the famous Theorem 94. ==Extension of classes== Let $K$ be an algebraic number field, called the ''base field'', and let $L/K$ be a field extension of finite degree. Let $\backslash mathcal\_K,\backslash mathcal\_K,\backslash mathcal\_K$ and $\backslash mathcal\_L,\backslash mathcal\_L,\backslash mathcal\_L$ denote the ring of integers, the group of nonzero fractional ideals and its subgroup of principal fractional ideals of the fields $K,L$ respectively. Then the extension map of frational ideals $\backslash iota\_:\backslash \; \backslash mathcal\_K\backslash to\backslash mathcal\_L,\backslash \; \backslash mathfrak\backslash mapsto\backslash mathfrak\backslash mathcal\_L$ is an injective group homomorphism. Since $\backslash iota\_(\backslash mathcal\_K)\backslash subseteq\backslash mathcal\_L$, this map induces the extension homomorphism of ideal class groups $j\_:\backslash \; \backslash mathcal\_K/\backslash mathcal\_K\backslash to\backslash mathcal\_L/\backslash mathcal\_L,\backslash \; \backslash mathfrak\backslash mathcal\_K\backslash mapsto(\backslash mathfrak\backslash mathcal\_L)\backslash mathcal\_L$. If there exists a non-principal ideal $\backslash mathfrak\backslash in\backslash mathcal\_K$ (i.e. $\backslash mathfrak\backslash mathcal\_K\backslash ne\backslash mathcal\_K$) whose extension ideal in $L$ is principal (i.e. $\backslash mathfrak\backslash mathcal\_L=A\backslash mathcal\_L$for some $A\backslash in\backslash mathcal\_L$ and $(\backslash mathfrak\backslash mathcal\_L)\backslash mathcal\_L=(A\backslash mathcal\_L)\backslash mathcal\_L=\backslash mathcal\_L$), then we speak about principalization or capitulation in $L/K$. In this case, the ideal $\backslash mathfrak$ and its class $\backslash mathfrak\backslash mathcal\_K$ are said to principalize or capitulate in $L$. This phenomenon is described most conveniently by the principalization kernel or capitulation kernel, that is the kernel $\backslash ker(j\_)$ of the class extension homomorphism. More generally, let $\backslash mathfrak=\backslash mathfrak\_0\backslash mathfrak\_\backslash infty$ be a modulus in $K$, where $\backslash mathfrak\_0$ is nonzero ideal in $\backslash mathcal\_K$ and $\backslash mathfrak\_\backslash infty$ is a formal product of pair-wise different real infinite primes of $K$. Then $\backslash mathcal\_\_K\backslash mid\backslash alpha\backslash equiv$ 1\pmod_K(\mathfrak) is the ray modulo $\backslash mathfrak$, where $\backslash mathcal\_K(\backslash mathfrak)=\backslash mathcal\_K(\backslash mathfrak\_0)$ is the group of nonzero fractional ideals in $K$ relatively prime to $\backslash mathfrak\_0$ and the condition $\backslash alpha\backslash equiv1\backslash pmod\_0\}$ and $v(\backslash alpha)>0$ for every real infinite prime $v$ dividing $\backslash mathfrak\_\backslash infty$. Let $\backslash mathcal\_\backslash le\backslash mathcal\_K(\backslash mathfrak)$, then the group $\backslash mathcal\_K(\backslash mathfrak)/\backslash mathcal$ is called generalized ideal class group for $\backslash mathfrak$. If $\backslash mathcal\_K(\backslash mathfrak\_K)/\backslash mathcal\_K$ and $\backslash mathcal\_L(\backslash mathfrak\_L)/\backslash mathcal\_L$ are generalized ideal class groups such that $\backslash mathfrak\backslash mathcal\_L\backslash in\backslash mathcal\_L(\backslash mathfrak\_L)$ for every $\backslash mathfrak\backslash in\backslash mathcal\_K(\backslash mathfrak\_K)$ and $\backslash mathfrak\backslash mathcal\_L\backslash in\backslash mathcal\_L$ for every $\backslash mathfrak\backslash in\backslash mathcal\_K$, then $\backslash iota\_$ induces the extension homomorphism of generalized ideal class groups $j\_:\backslash \; \backslash mathcal\_K(\backslash mathfrak\_K)/\backslash mathcal\_K\backslash to\backslash mathcal\_L(\backslash mathfrak\_L)/\backslash mathcal\_L,\backslash \; \backslash mathfrak\backslash mathcal\_K\backslash mapsto(\backslash mathfrak\backslash mathcal\_L)\backslash mathcal\_L$. 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「Principalization (algebra)」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. 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