THE DIFFERENT AND DIFFERENTIALS OF LOCAL-FIELDS WITH IMPERFECT RESIDUE FIELDS

Let K be a complete field with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension i s separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will ident ify the necessary correction term in the general case, and we give ine qualities for ramification groups of subextensions L'/K in terms of th ose of L/K. A question of Krasner in this context is settled with a co unterexample. These ramification phenomena can be related to the struc ture of the module of differentials of the extension of valuation ring s. For the case that [L : K] = p(2), where p is the residue characteri stic, this module is shown to determine the correction term in Hilbert 's formula.