# Frobenius theorem

A theorem that describes all finite-dimensional associative real algebras without divisors of zero; it was proved by G. Frobenius . Frobenius' theorem asserts that:

1) the field of real numbers and the field of complex numbers are the only finite-dimensional real associative-commutative algebras without divisors of zero; and

2) the skew-field of quaternions is the only finite-dimensional real associative, but not commutative, algebra without divisors of zero.

There is also a description of all finite-dimensional alternative algebras without divisors of zero:

3) the Cayley algebra is the only finite-dimensional real alternative, but not associative, algebra without divisors of zero.

The conjunction of these three assertions is called the generalized Frobenius theorem. All the algebras that appear in the statement of the theorem turn out to be algebras with unique division and with a one. The Frobenius theorem cannot be generalized to the case of non-alternative algebras. It has been proved, however, that the dimension of any finite-dimensional real algebra without divisors of zero can only take the values 1, 2, 4, or 8.

How to Cite This Entry:
Frobenius theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_theorem&oldid=46994
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article