ON THE CODIMENSION OF THE IMAGE OF A DERIVATION IN A COMMUTATIVE FIELD

Let delta 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 codimen sion, as k-vector space, of K delta + k in K is infinite, while in the infinitely generated case, the said codimension can be any finite non -negative integer.