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In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, on finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions or the octonions. The non-associative algebras occurring are called Hurwitz algebras or composition algebras. The problem has an equivalent formulation in terms of quadratic forms , composability requiring the existence of a bilinear "composition" such that . Subsequent proofs have used the Cayley–Dickson construction. Although neither commutative nor associative, composition algebras have the special property of being alternative algebras, i.e. left and right multiplication preserves squares, a weakened version of associativity. Their theory has subsequently been generalized to arbitrary quadratic forms and arbitrary fields.〔See: * * *〕 Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in . Subsequent proofs of the restrictions on the dimension have been given by using the representation theory of finite groups and by and using Clifford algebras. Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the classical groups〔See: * *〕 and in quantum mechanics to the classification of simple Jordan algebras. ==Euclidean Hurwitz algebras== 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「Hurwitz's theorem (composition algebras)」の詳細全文を読む スポンサード リンク
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