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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 Pfister form ： ウィキペディア英語版 Pfister form In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field ''F'' of characteristic not 2. For a natural number ''n'', an n-fold Pfister form over ''F'' is a quadratic form of dimension 2''n'' that can be written as a tensor product of quadratic forms :$\backslash langle\backslash !\backslash langle\; a\_1,\; a\_2,\; ...\; ,\; a\_n\; \backslash rangle\backslash !\backslash rangle\; \backslash cong\; \backslash langle\; 1,\; -a\_1\; \backslash rangle\; \backslash otimes\; \backslash langle\; 1,\; -a\_2\; \backslash rangle\; \backslash otimes\; ...\; \backslash otimes\; \backslash langle\; 1,\; -a\_n\; \backslash rangle,$ for some nonzero elements ''a''1, ..., ''a''''n'' of ''F''.〔 Elman, Karpenko, Merkurjev (2008), section 9.B.〕 (Some authors omit the signs in this definition; the notation here simplifies the relation to Milnor K-theory, discussed below.) An ''n''-fold Pfister form can also be constructed inductively from an (''n''–1)-fold Pfister form ''q'' and a nonzero element ''a'' of ''F'', as $q\; \backslash oplus\; (-a)q$. So the 1-fold and 2-fold Pfister forms look like: :$\backslash langle\backslash !\backslash langle\; a\backslash rangle\backslash !\backslash rangle\backslash cong\; \backslash langle\; 1,\; -a\; \backslash rangle\; =\; x^2\; -\; ay^2$. :$\backslash langle\backslash !\backslash langle\; a,b\backslash rangle\backslash !\backslash rangle\backslash cong\; \backslash langle\; 1,\; -a,\; -b,\; ab\; \backslash rangle\; =\; x^2\; -\; ay^2\; -\; bz^2\; +\; abw^2.$ For ''n'' ≤ 3, the ''n''-fold Pfister forms are norm forms of composition algebras.〔Lam (2005) p.316〕 In that case, two ''n''-fold Pfister forms are isomorphic if and only if the corresponding composition algebras are isomorphic. The ''n''-fold Pfister forms additively generate the ''n''-th power ''I''''n'' of the fundamental ideal of the Witt ring of ''F''.〔Lam (2005) p.316〕 ==Characterizations== A quadratic form ''q'' over a field ''F'' is multiplicative if, for vectors of indeterminates x and y, we can write ''q''(x).''q''(y) = ''q''(z) for some vector z of rational functions in the x and y over ''F''. Isotropic quadratic forms are multiplicative.〔Lam (2005) p.324〕 For anisotropic quadratic forms, Pfister forms are multiplicative, and conversely.〔Lam (2005) p.325〕 For ''n''-fold Pfister forms with ''n'' ≤ 3, this had been known since the 19th century; in that case ''z'' can be taken to be bilinear in ''x'' and ''y'', by the properties of composition algebras. It was a remarkable discovery by Pfister that ''n''-fold Pfister forms for all ''n'' are multiplicative in the more general sense here, involving rational functions. For example, he deduced that for any field ''F'' and any natural number ''n'', the set of sums of 2''n'' squares in ''F'' is closed under multiplication, using that the quadratic form $x\_1^2\; +\backslash cdots\; +\; x\_^2$ is an ''n''-fold Pfister form (namely, $\backslash langle\backslash !\backslash langle\; -1,\; ...\; ,\; -1\; \backslash rangle\backslash !\backslash rangle$).〔Lam (2005) p.319〕 Another striking feature of Pfister forms is that every isotropic Pfister form is in fact hyperbolic, that is, isomorphic to a direct sum of copies of the hyperbolic plane $\backslash langle\; 1,\; -1\; \backslash rangle$. This property also characterizes Pfister forms, as follows. If ''q'' is an anisotropic quadratic form over a field ''F'', and if ''q'' becomes hyperbolic over every extension field ''E'' such that ''q'' becomes isotropic over ''E'', then ''q'' is isomorphic to ''a''φ for some nonzero ''a'' in ''F'' and some Pfister form φ over ''F''.〔Elman, Karpenko, Merkurjev (2008), Corollary 23.4.〕 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「Pfister form」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.