Hurwitz's theorem (normed division algebras)

In algebra, Hurwitz's theorem (also called the “1,2,4 8 Theorem”), named after Adolf Hurwitz, who proved it in 1898, states: Every normed division algebra with an identity is isomorphic to one of the following four algebras: R, C, H and O, that is the real numbers, the complex numbers, the quaternions and the octonions.[1][2] The classification of real division algebras began with Georg Frobenius,[3] continued with Hurwitz[4] and was set in general form by Max Zorn.[5] A brief historical summary may be found in Badger.[6]

A full proof can be found in Kantor and Solodovnikov,[7] and in Shapiro.[8] As a basic idea, if an algebra A is proportional to 1 then it is isomorphic to the real numbers. Otherwise we extend the subalgebra isomorphic to 1 using the Cayley–Dickson construction and introducing a vector e which is orthogonal to 1. This subalgebra is isomorphic to the complex numbers. If this is not all of A then we once again use the Cayley–Dickson construction and another vector orthogonal to the complex numbers and get a subalgebra isomorphic to the quaternions. If this is not all of A then we double up once again and get a subalgebra isomorphic to the Cayley numbers (or Octonions). We now have a theorem which says that every subalgebra of A that contains 1 and is not A is associative. The Cayley numbers are not associative and therefore must be A.

Hurwitz's theorem can be used to prove that the product of the sum of n squares by the sum of n squares is the sum of n squares in a bilinear way only when n is equal to 1, 2, 4 and 8.[9]

In-line references

  1. ^ JA Nieto and LN Alejo-Armenta (2000). "Hurwitz theorem and parallelizable spheres from tensor analysis". Arxiv preprint hep-th/0005184. arXiv:hep-th/0005184. 
  2. ^ Kevin McCrimmon (2004). "Hurwitz's theorem 2.6.2". A taste of Jordan algebras. Springer. p. 166. ISBN 0387954473. http://books.google.com/books?id=6YG4ycpKMYkC&pg=PA166. "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." 
  3. ^ Georg Frobenius (1878). "Über lineare Substitutionen und bilineare Formen". J. Reine Angew. Math. 84: 1–63. 
  4. ^ Hurwitz, A. (1898). "Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln (On the composition of quadratic forms of arbitrary many variables)" (in German). Nachr. Ges. Wiss. Göttingen: 309–316. JFM 29.0177.01. 
  5. ^ Max Zorn (1930). "Theorie der alternativen Ringe". Abh. Math. Sem. Univ. Hamburg 8: 123–147. 
  6. ^ Matthew Badger. "Division algebras over the real numbers". http://www.math.washington.edu/~mbadger/divalg3.pdf. 
  7. ^ IL Kantor and AS Solodovnikov (1989). "Normed algebras with an identity. Hurwitz's theorem.". Hypercomplex numbers. An elementary introduction to algebras (2nd ed.). Springer-Verlag. p. 121. ISBN 0387969802. http://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=Normed+algebras+with+an+identity.+Hurwitz%27s+theorem&as_oq=&as_eq=&as_brr=0&as_pt=ALLTYPES&lr=&as_vt=&as_auth=&as_pub=&as_sub=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_isbn=&as_issn=. 
  8. ^ Daniel B. Shapiro (2000). "Appendix to Chapter 1. Composition algebras". Compositions of quadratic forms. Walter de Gruyter. pp. 21 ff. ISBN 311012629X. http://books.google.com/books?id=qrFhUda9JbkC&pg=PA21. 
  9. ^ Joe Roberts (1992). "Square identities". Lure of the integers. Cambridge University Press. ISBN 088385502X. http://books.google.com/books?id=DvX90EKMxGwC&pg=PA93. 

Background references