From: David Wilkins
Newsgroups: sci.math.research
Subject: Re: Extension of the complex domain
Date: 21 Feb 1996 17:45:39 -0000
In article <4gada8$39i@reader2.ix.netcom.com> Pierre1@ix.netcom.com (pierre touma) writes:
>
> A friend of mine who does not have access to the Internet is
>doing research on the extension of the complex domain beyond
>quaternions. The first part of his work is completed and he would like
>to have some feedback from interested researchers.
>
It seems appropriate to draw attention to the following textbook,
on the topic of extensions of the complex domain:
I.L. Kantor, A.S. Solodovnikov
Hypercomplex Numbers
An Elementary Introduction to Algebras
Translated by A. Shenitzer.
Springer-Verlag. New York/Berlin/Heidelberg. 1989
ISBN 0-387-96980-2
Included in this text are proofs of the following:
Hurwitz theorem. Every normed algebra with an identity is
isomorphic to one of the following four algebras: the real
numbers, the complex numbers, the quaternions, and the
Cayley numbers.
Frobenius's theorem. Every associative division algebra is
isomorphic to one of the following: the algebra of real numbers,
the algebra of complex numbers, and the algebra of quaternions.
The generalized Frobenius theorem. Every alternative division
algebra is isomorphic to one of the following four algebras:
the real numbers, the complex numbers, the quaternions, and
the Cayley numbers.