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