On the codimension of the image of a derivation in a communicative field

Abstract

Let δ be a derivation in a field K of characteristic 0, with fixed field k. We show that, if K/k is finitely generated, then the codimension, as k-vector space, of K δ + k in K is infinite, while in the infinitely generated case, the said codimension can be any finite non-negative integer.