<< There are lots of infinite dimensional fields over the real or complex numbers, but what about infinite dimensional noncommutative or nonassociative division algebras? What is known about such creatures?
Years ago when Conway was active here, he stated that there are no infinite-dimensional real division algebras, unless I'm remembering wrong. That would seem to rule out complex ones as well, would you agree? --Dan ________________________________ Well there are lots of infinite dimensional fields, e.g. function fields, F extended by one or more transcendentals. Are you you saying that there is a Wedderburn type theorem here, that all infinite dimensional (real or complex) division algebras are fields? Is there such an animal as a free division algebra?
Division algebras as I use the term can be commutative or non. By an infinite-dimensional real division algebra I mean a multiplication on the structure of an infinite-dimensional real vector space like R^(oo) (all sequences of reals with only finitely many nonzero terms), with the usual added axioms. (Actually I don't recall if Conway said anything about dimensions > aleph_0.) Hmm, let's see. Is there a way to show that the field of rational functions in X over the reals with its addition and scalar multiplication is a real vector space isomorphic to R^(oo) ? It certainly seems so (even if it requires AC to get a basis). Then field multiplication sure seems to give a division algebra structure. Am I missing something? (It may be significant that neither plain googling, nor Google Scholar, nor MathSciNet gives any hits at all for the phrase "infinite-dimensional real division algebra", with or without the hyphen. But even if they don't exist, I'd expect some math paper to say so.) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele