Jump to content

Artin–Zorn theorem

From Wikipedia, the free encyclopedia

In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Artin.[1][2]

The Artin–Zorn theorem is a generalization of the Wedderburn theorem, which states that finite associative division rings are fields. As a geometric consequence, every finite Moufang plane is the classical projective plane over a finite field.[3][4]

References

[edit]
  1. ^ Zorn, M. (1930), "Theorie der alternativen Ringe", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 8: 123–147, doi:10.1007/BF02940993, S2CID 121384721.
  2. ^ Lüneburg, Heinz (2001), "On the early history of Galois fields", in Jungnickel, Dieter; Niederreiter, Harald (eds.), Finite fields and applications: proceedings of the Fifth International Conference on Finite Fields and Applications Fq5, held at the University of Augsburg, Germany, August 2–6, 1999, Springer-Verlag, pp. 341–355, ISBN 978-3-540-41109-3, MR 1849100.
  3. ^ Shult, Ernest (2011), Points and Lines: Characterizing the Classical Geometries, Universitext, Springer-Verlag, p. 123, ISBN 978-3-642-15626-7.
  4. ^ McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Springer-Verlag, p. 34, ISBN 978-0-387-95447-9.