# Explicit integral Galois module structure of weakly ramified extensions of local fields – Henri Johnston (Univiesity of Exeter)

#### Abstract

Let $L/K$ be a finite Galois extension of complete local fields with finite residue fields and let $G={\rm Gal}(L/K)$. 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 $\mathcal{O}_{L}$ be the valuation ring of $L$ and let $\mathfrak{P}_{L}$ be its maximal ideal. We show that if $L/K$ is weakly ramified and $n \equiv 1 \bmodG_{1}$ then $\mathfrak{P}_{L}^{n}$ is free over the group ring $\mathcal{O}_{K}[G]$, 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 $\mathfrak{P}_{L}$ over $\mathcal{O}_{K}[G]$ is also a free generator of $\mathcal{O}_{L}$ over its associated order in the group algebra $K[G]$

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