Gene wrote: << --- dasimov@earthlink.net wrote: ...
* Real division algebras occur only in dimensions 1,2,4,8. ...
You mean to say: Finite dimensional real division algebras. There are lots of infinite dimensional real division algebras. Some examples are the fields of rational functions R(x), R(x,y), etc.
Thanks for pointing that out -- it hadn't occurred to me. (Plus, I seem to recall JHC's stating in this venue that there exist no infinite-dimensional real division algebras.) But I see no problem with R(x), etc., as real division algebras. Which led me to wonder what a basis might be for R(x) as a real vector space. I suppose one possible basis for R(x) is { x^k / P(x)^n : P(x) a real monic poly. irreducible* over R with k, n nonnegative}. *I.e., P(x) is of form x-c (c real), or else x^2 + bx + c (b,c real with b^2 < 4c). This implies R(x) has dimension 2^aleph_0 as a vector space over R. Could JHC have been referring only to the nonexistence of real division algebras of dimension aleph_0 ? --Dan