On Galois isomorphisms between ideals in extensions of local fields

Abstract

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.