In
mathematics, the Artin–Zorn theorem, named after
Emil Artin and
Max Zorn
Max August Zorn (; June 6, 1906 – March 9, 1993) was a German mathematician. He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn's lemma, a method used in set theory that is applicable to a wide range of ...
, states that any finite
alternative division ring In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
*x(xy) = (xx)y
*(yx)x = y(xx)
for all ''x'' and ''y'' in the algebra.
Every associative algebra is ...
is necessarily a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Artin.
The Artin–Zorn theorem is a generalization of the
Wedderburn theorem In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields.
The Artin–Zorn theorem generalizes the theorem to alter ...
, which states that finite associative division rings are fields. As a geometric consequence, every finite
Moufang plane
In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective plane that has a ''translation line'', that is, a line with the proper ...
is the classical projective plane over a finite field.
[.]
References
Theorems in ring theory
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