Yes, and indeed the field of rational functions in any finite or infinite number of variables. -- Gene On Friday, November 16, 2018, 12:44:26 PM PST, Dan Asimov <dasimov@earthlink.net> wrote: All this talk of polynomials suggested something I hadn't noticed before: Let P(X) = Sum_{1 <= k <= n} a_n X^n for any (say) reals a_k. Then think of P(X) as the point (a_0, a_1, ..., a_n) in R^(n+1). Polynomials add just like vectors in R^(n+1). Using this we can also multiply vectors v in R^(n+1) and w in R(m+1) to get one in R^(n+m+1). So if we take the union of all R^n, 1 <= n < oo (call this R^oo), we get a (commutative, associative) ring with unit. In fact it's an integral domain (the product of nonzero elements is nonzero). Hence R^oo has a field of fractions, which I suppose is equivalent to the field of rational functions over R. It's even a topological field, since addition and multiplication are continuous functions of the coefficients. Is this not a real division algebra? I've read that Kaplansky and others proved that the only real division algebras are in 1, 2, 4, and 8 dimensions. In particular, there are *none* in infinitely many dimensions (excepting the case of vector spaces over purely inseparable fields, which I don't think applies to the reals). Whatever it is, it's a pretty cool object. —Dan