On Galois isomorphisms between ideals in extensions of local fields

LetL/K be a totally ramified, finite abelian extension of local fields, let\(\mathfrak{O}_L \) and\(\mathfrak{O}\) be the valuation rings, and letG be the Galois group. We consider the powers\(\mathfrak{P}_L^r \) of the maximal ideal of\(\mathfrak{O}_L \) as modules over the group ring\(\mathfrak{O}G\). We show that, ifG has orderp m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then\(\mathfrak{P}_L^r \) and\(\mathfrak{P}_L^{r'} \) are isomorphic iffr≡r′ modp m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.