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 $K[G]$ <\/p>\n","protected":false},"excerpt":{"rendered":"
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…<\/p>\n