On Galois isomorphisms between ideals in extensions of local fields (Q1191485)

Let \(L/K\) be a totally ramified, finite abelian extension of local fields, let \({\mathfrak O}_ L\) and \({\mathfrak O}\) be the valuation rings, and let \(G\) be the Galois group. The powers \({\mathfrak P}_ L^ r\) of the maximal ideal of \({\mathfrak O}_ L\) are considered as modules over the group ring \({\mathfrak O}G\). It is shown 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 condition on the ramification of \(L/K\) holds, then \({\mathfrak P}_ L^ r\) and \({\mathfrak P}_ L^{r'}\) are isomorphic iff \(r\equiv r'\bmod p^ m\). Also a generalization of this result to certain extensions not of \(p\)- power degree is given, and it is shown that, in the case \(p=2\), the hypotheses that \(G\) is abelian and not cyclic can be removed. The proof of these results involves an analysis of the case where \(L/K\) is cyclic of degree \(p\) and a reduction of the general case to this special case using the notion of factor equivalence.