Explicit integral Galois module structure of weakly ramified extensions of local fields

TL;DR

This paper investigates the Galois module structure of ramified extensions of local fields, showing conditions under which valuation rings are free over group rings and constructing explicit generators.

Contribution

It provides explicit constructions of free generators for valuation rings in weakly ramified extensions and extends results to wildly ramified cases, including a new splitting lemma.

Findings

P_L^n is free over O_K[G] when n ≡ 1 mod |G_1| in weakly ramified extensions.

Explicit generators for valuation rings are constructed.

Every free generator over O_K[G] is also a free generator over the associated order in the wildly ramified case.

Abstract

Let L/K be a finite Galois extension of complete local fields with finite residue fields and let G=Gal(L/K). Let G_1 and G_2 be the first and second ramification groups. Thus L/K is tamely ramified when G_1 is trivial and we say that L/K is weakly ramified when G_2 is trivial. Let O_L be the valuation ring of L and let P_L be its maximal ideal. We show that if L/K is weakly ramified and n is congruent to 1 mod |G_1| then P_L^n is free over the group ring O_K[G], and we construct an explicit generating element. Under the additional assumption that L/K is wildly ramified, we then show that every free generator of P_L over O_K[G] is also a free generator of O_L over its associated order in the group algebra K[G]. Along the way, we prove a `splitting lemma' for local fields, which may be of independent interest.