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In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form which satisfies : for all and in . Unital composition algebras are called Hurwitz algebras.〔Okubo (1995) p. 22〕 If the ground field is the field of real numbers and is positive-definite, then is called an Euclidean Hurwitz algebra. The quadratic form is often referred to as a ''norm'' on . Composition algebras are also called normed algebras: these should not be confused with associative normed algebras, which include Banach algebras, although three associative Euclidean Hurwitz algebras , , and in fact ''are'' Banach algebras. ==Structure theorem== Every unital composition algebra over a field can be obtained by repeated application of the Cayley–Dickson construction starting from (if the characteristic of is different from 2) or a 2-dimensional composition subalgebra (if ). The possible dimensions of a composition algebra are 1, 2, 4, and 8.〔Jacobson (1958); Roos (2008); Schafer (1995) p. 73〕 *1-dimensional composition algebras only exist when . *Composition algebras of dimension 1 and 2 are commutative and associative. *Composition algebras of dimension 2 are either quadratic field extensions of or isomorphic to . *Composition algebras of dimension 4 are called quaternion algebras. They are associative but not commutative. *Composition algebras of dimension 8 are called octonion algebras. They are neither associative nor commutative. 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「Composition algebra」の詳細全文を読む スポンサード リンク
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