Artin–Zorn theorem

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]


  1. Zorn, M. (1930), "Theorie der alternativen Ringe", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 8: 123–147.
  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.

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