翻訳と辞書 Words near each other ・ Hilbert's nineteenth problem ・ Hilbert's ninth problem ・ Hilbert's Nullstellensatz ・ Hilbert's paradox of the Grand Hotel ・ Hilbert's problems ・ Hilbert's program ・ Hilbert's second problem ・ Hilbert's seventeenth problem ・ Hilbert's seventh problem ・ Hilbert's sixteenth problem ・ Hilbert's sixth problem ・ Hilbert's syzygy theorem ・ Hilbert's tenth problem ・ Hilbert's theorem ・ Hilbert's theorem (differential geometry) ・ Hilbert's Theorem 90 ・ Hilbert's third problem ・ Hilbert's thirteenth problem ・ Hilbert's twelfth problem ・ Hilbert's twentieth problem ・ Hilbert's twenty-first problem ・ Hilbert's twenty-fourth problem ・ Hilbert's twenty-second problem ・ Hilbert's twenty-third problem ・ Hilbert, West Virginia ・ Hilbert, Wisconsin ・ Hilbert–Bernays paradox ・ Hilbert–Bernays provability conditions ・ Hilbert–Burch theorem ・ Hilbert–Huang transform 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 Hilbert's Theorem 90 ： ウィキペディア英語版 Hilbert's Theorem 90 In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if ''L''/''K'' is a cyclic extension of fields with Galois group ''G'' = Gal(''L''/''K'') generated by an element s and if a is an element of ''L'' of relative norm 1, then there exists b in ''L'' such that : ''a'' = s(''b'')/''b''. The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's famous Zahlbericht , although it is originally due to . Often a more general theorem due to is given the name, stating that if ''L''/''K'' is a finite Galois extension of fields with Galois group ''G'' = Gal(''L''/''K''), then the first cohomology group is trivial: :''H''1(''G'', ''L''×) = == Examples == Let ''L''/''K'' be the quadratic extension $\backslash mathbb(i)\; /\; \backslash mathbb$. The Galois group is cyclic of order 2, its generator s acting via conjugation: :$s:\backslash ,\; \backslash ,\; c\; -\; di\backslash mapsto\; c\; +\; di\backslash \; .$ An element $x=a+bi$ in ''L'' has norm $xx^=a^2\; +b^2$. An element of norm one corresponds to a rational solution of the equation $a^2+b^2=1$ or in other words, a point with rational coordinates on the unit circle. Hilbert's Theorem 90 then states that every element ''y'' of norm one can be parametrized (with integral ''c'', ''d'') as :$y=\backslash frac=\backslash frac\; +\; \backslash frac\; i$ which may be viewed as a rational parametrization of the rational points on the unit circle. Rational points $\backslash ,\; (x,y)=(a/c,b/c)$ on the unit circle $x^2+y^2=1$ correspond to Pythagorean triples, i.e. triples $\backslash ,(a,b,c)$ of integers satisfying $\backslash ,\; a^2+b^2=c^2$. 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「Hilbert's Theorem 90」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.