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In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for 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. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras.

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  • Théorème de Frobenius généralisé
  • Теорема Гурвица о нормированных алгебрах с делением
  • 胡尔维兹定理
  • Hurwitz's theorem (composition algebras)
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  • En mathématiques, diverses versions de théorèmes de Frobenius généralisés ont étendu progressivement le théorème de Frobenius de 1877. Ce sont des théorèmes d'algèbre générale qui classifient les algèbres unifères à division de dimension finie sur le corps commutatif ℝ des réels. Moyennant certaines restrictions, il n'y en a que quatre : ℝ lui-même, ℂ (complexes), ℍ (quaternions) et 𝕆 (octonions).
  • 在代数学中,胡尔维兹定理(又名“1,2,4,8定理”)是以在1898年证明它的阿道夫·胡尔维兹命名。该定理表明:任何带有单位元的赋范可除代数同构于以下四个代数之一:R,C,H和O,分别代表实数、复数、四元数和八元数。对实赋范可除代数的分类始于弗洛比纽斯 ,发扬于胡尔维兹,由佐恩整理为一般形式。一个简短的历史摘要可见Badger。 完整的证明能在凯特和索洛多斯尼科夫或者夏皮罗处找到。一个基本的想法是,如果一个代数A是成正比于1的,那么它同构于实数。否则,我们使用凯莱-迪克森结构扩展子代数以同构于1,并引入一个向量正交于1。此子代数是同构于复数的。如果它不是A的全体,那么我们再次使用凯莱-迪克森结构和另一个与复数正交的向量,得到一个与四元数同构的子代数。如果这还不是不是A的全体,我们重复以上行为一次,并得到同构于凯莱数(或八元数)的子代数。我们现在有一个定理,说的是每一个包含1而又不是A自身的子代数是结合的。凯莱数不是结合的,因此必须为A。 胡尔维兹定理也可以用于证明n个平方和与n个平方和的积仍可以写成n个平方和仅当n为1,2,4或者8时。
  • Теорема Гурвица о нормированных алгебрах — утверждение о множестве всех возможных алгебр с единицей, допускающих при введении скалярного произведения правило «норма произведения равна произведению норм» (нормированная алгебра). Установлена немецким математиком Гурвицем в 1898 году..
  • In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for 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. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras.
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