ON GALOIS ISOMORPHISMS BETWEEN IDEALS IN EXTENSIONS OF LOCAL-FIELDS

Let L/K be a totally ramified, finite abelian extension of local fields, let D(L) and D be the valuation rings, and let G be the Galois group. We consider the powers B(L)r of the maximal ideal of D(L) as modules over the group ring DG. We show that, if G has order p(m) (with p the residue field characteristic), if G is not cyclic (or if G has order p), and if a certain mild hypothesis on the ramification of L/K holds, then B(L)r and B(L)r' are isomorphic iff r = r' mod p(m). We also give a generalisation of this result to certain extensions not of p-power degree, and show that, in the case p = 2, the hypotheses that G is abelian and not cyclic can be removed.