胡尔维兹定理

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：任何带有单位元的賦範可除代數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。^[1]^[2]对实賦範可除代數的分类始于弗洛比纽斯^[3] ，发扬于胡尔维兹^[4]，由佐恩整理为一般形式^[5]。一个简短的历史摘要可见Badger^[6]。

完整的证明能在凯特和索洛多斯尼科夫^[7]或者夏皮罗^[8]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^[9]。

参考文献

引用

^ JA Nieto and LN Alejo-Armenta. Hurwitz theorem and parallelizable spheres from tensor analysis. Arxiv preprint hep-th/0005184. 2000 [2011-01-04]. （原始内容存档于2022-04-14）.
^ Kevin McCrimmon. Hurwitz's theorem 2.6.2. A taste of Jordan algebras. Springer. 2004: 166. ISBN 0387954473. Only recently was it established that the only finite-dimensional real nonassociative division algebras have dimensions 1,2,4,8; the algebras were not classified, and the proof was topological rather than algebraic.
^ Georg Frobenius. Über lineare Substitutionen und blineare Formen. J. Reine Angew. Math. 1878, 84: 1–63.
^ Hurwitz, A. Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln (On the composition of quadratic forms of arbitrary many variables). Nachr. Ges. Wiss. Göttingen. 1898: 309–316. Template:JFM （德语）.
^ Max Zorn. Theorie der alternativen Ringe. Abh. Math. Sem. Univ. Hamburg. 1930, 8: 123–147.
^ Matthew Badger. Division algebras over the real numbers (PDF). （原始内容 (PDF)存档于2011-06-07）.
^ IL Kantor and AS Solodovnikov. Normed algebras with an identity. Hurwitz's theorem.. Hypercomplex numbers. An elementary introduction to algebras 2nd. Springer-Verlag. 1989: 121. ISBN 0387969802.
^ Daniel B. Shapiro. Appendix to Chapter 1. Composition algebras. Compositions of quadratic forms. Walter de Gruyter. 2000: 21 ff. ISBN 311012629X.
^ Joe Roberts. Square identities. Lure of the integers. Cambridge University Press. 1992. ISBN 088385502X.

书籍

John H. Conway, Derek A. Smith On Quaternions and Octonions. A.K. Peters, 2003.
John Baez, The Octonions, AMS 2001.

[Nieto-1] JA Nieto and LN Alejo-Armenta. Hurwitz theorem and parallelizable spheres from tensor analysis. Arxiv preprint hep-th/0005184. 2000 [2011-01-04]. （原始内容存档于2022-04-14）.

[McCrimmon-2] Kevin McCrimmon. Hurwitz's theorem 2.6.2. A taste of Jordan algebras. Springer. 2004: 166. ISBN 0387954473. Only recently was it established that the only finite-dimensional real nonassociative division algebras have dimensions 1,2,4,8; the algebras were not classified, and the proof was topological rather than algebraic.

[Frobenius-3] Georg Frobenius. Über lineare Substitutionen und blineare Formen. J. Reine Angew. Math. 1878, 84: 1–63.

[Hurwitz-4] Hurwitz, A. Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln (On the composition of quadratic forms of arbitrary many variables). Nachr. Ges. Wiss. Göttingen. 1898: 309–316. Template:JFM （德语）.

[Zorn-5] Max Zorn. Theorie der alternativen Ringe. Abh. Math. Sem. Univ. Hamburg. 1930, 8: 123–147.

[Badger-6] Matthew Badger. Division algebras over the real numbers (PDF). （原始内容 (PDF)存档于2011-06-07）.

[Kantor-7] IL Kantor and AS Solodovnikov. Normed algebras with an identity. Hurwitz's theorem.. Hypercomplex numbers. An elementary introduction to algebras 2nd. Springer-Verlag. 1989: 121. ISBN 0387969802.

[Shapiro-8] Daniel B. Shapiro. Appendix to Chapter 1. Composition algebras. Compositions of quadratic forms. Walter de Gruyter. 2000: 21 ff. ISBN 311012629X.

[squares-9] Joe Roberts. Square identities. Lure of the integers. Cambridge University Press. 1992. ISBN 088385502X.

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胡尔维兹定理

参考文献

引用

书籍

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