Genus change in inseparable extensions of function fields

Abstract

holds for all tGK. Sa is a particularly convenient substitute for the trace from K to k, which is identically 0. Of course Sa, although not completely arbitrary, is nevertheless noninvariant, and the question arises as to how Sa transforms if we replace a by another generator ,B. This question can be more precisely stated if we recall that since K is a field and Sa is nontrivial, any k-linear map S of K into k can be expressed in the form S(t) =Sa(,,y), where y is some element of K uniquely determined by S. Our question is therefore: How does one compute, in terms of a and ,B, the element -y for which So(t) = S.(tzy) ? The answer is most conveniently expressed in terms of derivations. A derivation in a ring is a map x-?Dx of the ring into itself with the properties D(x+y) =D(x)+D(y) and D(xy) =x(Dy)+(Dx)y. The rule D(xr) =vx'-'Dx follows by induction if the ring is commutative. The ordinary formal differentiation F(X)--F'(X) is a derivation in the ring k [X] of polynomials in one letter X over our field k. It maps a principal ideal generated by a polynomial of the form XP-a into itself because ((XP a) F(X)) '(XP-a) F'(X). The