翻訳と辞書 Words near each other ・ Arganarhinus ・ Arganasaurus ・ Arganasuchus ・ Arganchy ・ Argancy ・ Argand ・ Argand lamp ・ Argand system ・ Arganda del Rey ・ Arganda del Rey (Madrid Metro) ・ Arezzo Wave ・ Areíto ・ Areíto (album) ・ ARF ・ ARF History Museum ・ Arf invariant ・ Arf invariant of a knot ・ Arf ring ・ Arf semigroup ・ ARF1 ・ ARF3 ・ ARF4 ・ ARF5 ・ ARF6 ・ Arfa ・ Arfa Deh ・ Arfa Karim ・ Arfa Software Technology Park ・ Arfaja al-Bariqi ・ Arfak astrapia 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 Arf invariant ： ウィキペディア英語版 Arf invariant In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, for the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2. In the special case of the 2-element field F2 the Arf invariant can be described as the element of F2 that occurs most often among the values of the form. Two nonsingular quadratic forms over F2 are isomorphic if and only if they have the same dimension and the same Arf invariant. This fact was essentially known to , even for any finite field of characteristic 2, and Arf proved it for an arbitrary perfect field. An assessment of Arf's results in the framework of the theory of quadratic forms can be found in.〔F. Lorenz and P. Roquette. Cahit Arf and his invariant.〕 The Arf invariant is particularly applied in geometric topology, where it is primarily used to define an invariant of -dimensional manifolds (singly even-dimensional manifolds: surfaces (2-manifolds), 6-manifolds, 10-manifolds, etc.) with certain additional structure called a framing, and thus the Arf–Kervaire invariant and the Arf invariant of a knot. The Arf invariant is analogous to the signature of a manifold, which is defined for 4''k''-dimensional manifolds (doubly even-dimensional); this 4-fold periodicity corresponds to the 4-fold periodicity of L-theory. The Arf invariant can also be defined more generally for certain 2''k''-dimensional manifolds. ==Definitions== The Arf invariant is defined for a quadratic form ''q'' over a field ''K'' of characteristic 2 such that ''q'' is nonsingular, in the sense that the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$ is nondegenerate. The form $b$ is alternating since ''K'' has characteristic 2; it follows that a nonsingular quadratic form in characteristic 2 must have even dimension. Any binary (2-dimensional) nonsingular quadratic form over ''K'' is equivalent to a form $q(x,y)=\; ax^2\; +\; xy\; +\; by^2$ with $a,\; b$ in ''K''. The Arf invariant is defined to be the product $ab$. If the form $q\text{'}(x,y)=a\text{'}x^2\; +\; xy+b\text{'}y^2$ is equivalent to $q(x,y)$, then the products $ab$ and $a\text{'}b\text{'}$ differ by an element of the form $u^2+u$ with $u$ in ''K''. These elements form an additive subgroup ''U'' of ''K''. Hence the coset of $ab$ modulo U is an invariant of $q$, which means that it is not changed when $q$ is replaced by an equivalent form. Every nonsingular quadratic form $q$ over ''K'' is equivalent to a direct sum $q\; =\; q\_1\; +\; \backslash ldots\; +\; q\_r$ of nonsingular binary forms. This was shown by Arf, but it had been earlier observed by Dickson in the case of finite fields of characteristic 2. The Arf invariant Arf($q$) is defined to be the sum of the Arf invariants of the $q\_i$. By definition, this is a coset of K modulo U. Arf〔Arf (1941)〕 showed that indeed Arf($q$) does not change if $q$ is replaced by an equivalent quadratic form, which is to say that it is an invariant of $q$. The Arf invariant is additive; in other words, the Arf invariant of an orthogonal sum of two quadratic forms is the sum of their Arf invariants. For a field ''K'' of characteristic 2, Artin-Schreier theory identifies the quotient group of ''K'' by the subgroup ''U'' above with the Galois cohomology group ''H''1(''K'', F2). In other words, the nonzero elements of ''K''/''U'' are in one-to-one correspondence with the separable quadratic extension fields of ''K''. So the Arf invariant of a nonsingular quadratic form over ''K'' is either zero or it describes a separable quadratic extension field of ''K''. This is analogous to the discriminant of a nonsingular quadratic form over a field ''F'' of characteristic not 2. In that case, the discriminant takes values in ''F'' */(''F'' *)2, which can be identified with ''H''1(''F'', F2) by Kummer theory. 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「Arf invariant」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.