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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 Leopoldt's conjecture ： ウィキペディア英語版 Leopoldt's conjecture In algebraic number theory, Leopoldt's conjecture, introduced by , states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual regulator defined using p-adic logarithms instead of the usual logarithms, introduced by . Leopoldt proposed a definition of a p-adic regulator ''R''''p'' attached to ''K'' and a prime number ''p''. The definition of ''R''''p'' uses an appropriate determinant with entries the p-adic logarithm of a generating set of units of ''K'' (up to torsion), in the manner of the usual regulator. The conjecture, which for general ''K'' is still open , then comes out as the statement that ''R''''p'' is not zero. ==Formulation== Let ''K'' be a number field and for each prime ''P'' of ''K'' above some fixed rational prime ''p'', let ''U''''P'' denote the local units at ''P'' and let ''U''1,''P'' denote the subgroup of principal units in ''U''''P''. Set : $U\_1\; =\; \backslash prod\_\; U\_.$ Then let ''E''1 denote the set of global units ''ε'' that map to ''U''1 via the diagonal embedding of the global units in ''E''. Since $E\_1$ is a finite-index subgroup of the global units, it is an abelian group of rank $r\_1\; +\; r\_2\; -\; 1$, where $r\_1$ is the number of real embeddings of $K$ and $r\_2$ the number of pairs of complex embeddings. Leopoldt's conjecture states that the $\backslash mathbb\_p$-module rank of the closure of $E\_1$ embedded diagonally in $U\_1$ is also $r\_1\; +\; r\_2\; -\; 1.$ Leopoldt's conjecture is known in the special case where $K$ is an abelian extension of $\backslash mathbb$ or an abelian extension of an imaginary quadratic number field: reduced the abelian case to a p-adic version of Baker's theorem, which was proved shortly afterwards by . has announced a proof of Leopoldt's conjecture for all CM-extensions of $\backslash mathbb$. expressed the residue of the ''p''-adic Dedekind zeta function of a totally real field at ''s'' = 1 in terms of the ''p''-adic regulator. As a consequence, Leopoldt's conjecture for those fields is equivalent to their ''p''-adic Dedekind zeta functions having a simple pole at ''s'' = 1. 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「Leopoldt's conjecture」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.