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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 Reflection theorem ： ウィキペディア英語版 Reflection theorem :''For reflection principles in set theory, see reflection principle.'' In algebraic number theory, a reflection theorem or Spiegelungssatz (German for ''reflection theorem'' – see ''Spiegel'' and ''Satz'') is one of a collection of theorems linking the sizes of different ideal class groups (or ray class groups), or the sizes of different isotypic components of a class group. The original example is due to Ernst Eduard Kummer, who showed that the class number of the cyclotomic field $\backslash mathbb\; \backslash left(\; \backslash zeta\_p\; \backslash right)$, with ''p'' a prime number, will be divisible by ''p'' if the class number of the maximal real subfield $\backslash mathbb\; \backslash left(\; \backslash zeta\_p\; \backslash right)^$ is. Another example is due to Scholz.〔A. Scholz, Uber die Beziehung der Klassenzahlen quadratischer Korper zueinander, ''J. reine angew. Math.'', 166 (1932), 201-203.〕 A simplified version of his theorem states that if 3 divides the class number of a real quadratic field $\backslash mathbb\; \backslash left(\; \backslash sqrt\; \backslash right)$, then 3 also divides the class number of the imaginary quadratic field $\backslash mathbb\; \backslash left(\; \backslash sqrt\; \backslash right)$. ==Leopoldt's Spiegelungssatz== Both of the above results are generalized by Leopoldt's "Spiegelungssatz", which relates the p-ranks of different isotypic components of the class group of a number field considered as a module over the Galois group of a Galois extension. Let ''L''/''K'' be a finite Galois extension of number fields, with group ''G'', degree prime to ''p'' and ''L'' containing the ''p''-th roots of unity. Let ''A'' be the ''p''-Sylow subgroup of the class group of ''L''. Let φ run over the irreducible characters of the group ring Q''p''() and let ''A''φ denote the corresponding direct summands of ''A''. For any φ let ''q'' = ''p''φ(1) and let the ''G''-rank ''e''φ be the exponent in the index :$(A\_\backslash phi\; :\; A\_\backslash phi^p)\; =\; q^\; .$ Let ω be the character of ''G'' :$\backslash zeta^g\; =\; \backslash zeta^\; \backslash text\; \backslash zeta\; \backslash in\; \backslash mu\_p\; .$ The reflection (''Spiegelung'') φ * is defined by :$\backslash phi^$ *(g) = \omega(g) \phi(g^) . Let ''E'' be the unit group of ''K''. We say that ε is "primary" if $K(\backslash sqrt()\backslash epsilon)/K$ is unramified, and let ''E''0 denote the group of primary units modulo ''E''''p''. Let δφ denote the ''G''-rank of the φ component of ''E''0. The Spiegelungssatz states that :$|\; e\_\; -\; e\_\backslash phi\; |\; \backslash le\; \backslash delta\_\backslash phi\; .$ 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「Reflection theorem」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.