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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 Quaternion algebra ： ウィキペディア英語版 Quaternion algebra In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''〔See Peirce. Associative algebras. Springer. Lemma at page 14.〕〔See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2.〕 that has dimension 4 over ''F''. Every quaternion algebra becomes the matrix algebra by ''extending scalars'' (=tensoring with a field extension), i.e. for a suitable field extension ''K'' of ''F'', $A\; \backslash otimes\_F\; K$ is isomorphic to the 2×2 matrix algebra over ''K''. The notion of a quaternion algebra can be seen as a generalization of the Hamilton quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over $F\; =\; \backslash mathbb$ (the real number field), and indeed the only one over $\backslash mathbb$ apart from the 2×2 real matrix algebra, up to isomorphism. ==Structure== ''Quaternion algebra'' here means something more general than the algebra of Hamilton quaternions. When the coefficient field ''F'' does not have characteristic 2, every quaternion algebra over ''F'' can be described as a 4-dimensional ''F''-vector space with basis $\backslash$, with the following multiplication rules: :$i^2=a$ :$j^2=b$ :$ij=k$ :$ji=-k$ where ''a'' and ''b'' are any given nonzero elements of ''F''. From these rules we get: :$k^2=ijij=-iijj=-ab$ The classical instances where $F=\backslash mathbb$ are Hamilton quaternions (''a'' = ''b'' = −1) and split-quaternions (''a'' = −1, ''b'' = +1). In split-quaternions, $k^2\; =\; +1\; \backslash \; \backslash and\; \backslash \; j\; k\; =\; -\; i\; \backslash \; ,$, contrary to Hamilton's equations. The algebra defined in this way is denoted (''a'',''b'')''F'' or simply (''a'',''b'').〔Gille & Szamuely (2006) p.2〕 When ''F'' has characteristic 2, a different explicit description in terms of a basis of 4 elements is also possible, but in any event the definition of a quaternion algebra over ''F'' as a 4-dimensional central simple algebra over ''F'' applies uniformly in all characteristics. A quaternion algebra (''a'',''b'')''F'' is either a division algebra or isomorphic to the matrix algebra of 2×2 matrices over ''F'': the latter case is termed ''split''.〔Gille & Szamuely (2006) p.3〕 The ''norm form'' :$N(t\; +\; xi\; +yj\; +\; zk)\; =\; t^2\; -\; ax^2\; -\; by^2\; +\; abz^2\; \backslash$ defines a structure of division algebra if and only if the norm is an anisotropic quadratic form, that is, zero only on the zero element. The conic ''C''(''a'',''b'') defined by :$a\; x^2\; +\; b\; y^2\; =\; z^2\; \backslash$ has a point (''x'',''y'',''z'') with coordinates in ''F'' in the split case.〔Gille & Szamuely (2006) p.7〕 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「Quaternion algebra」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.