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