From: hrubin@b.stat.purdue.edu (Herman Rubin)
Newsgroups: sci.math
Subject: Re: Division algebras over real algebraic numbers
Date: 15 Jan 1996 11:50:06 -0500
In article ,
Laura Helen wrote:
>Are there finite dimensional division algebras over the real
>algebraic numbers which can't be embedded
>in a division algebra over R of the same dimension?
>For example, the quaternions with real algebraic-number coefficients are
>a division algebra over the real algebraic numbers.
>Such a division algebra couldn't have odd dimension. Could it be
>shown that it can't have dimensions other than 1,2,4,8 by topological
>arguments similar to those used to prove this for division algebras
>over the reals?
As Tarski showed, all real closed fields are elemantarily equivalent.
Having a division algebra of a given finite dimension over a field
being an elementary property, this provides a proof.
Topological arguments cannot be used; it is the case that an
algebraic function attains its maximum in an algebraic interval,
but not an arbitrary continuous function.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558