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

Author: Henri Johnston
Journal: Proc. Amer. Math. Soc. 143 (2015), 5059-5071
MSC (2010): Primary 11R33, 11S15
Published electronically: May 7, 2015
MathSciNet review: 3411126
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Abstract: Let $ L/K$ be a finite Galois extension of complete local fields with finite residue fields and let $ G=\operatorname {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 \bmod \vert G_{1}\vert$, 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]$. Along the way, we prove a `splitting lemma' for local fields, which may be of independent interest.

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Henri Johnston
Affiliation: Department of Mathematics, University of Exeter, Exeter, EX4 4QF, United Kingdom

Received by editor(s): August 20, 2014
Received by editor(s) in revised form: September 15, 2014
Published electronically: May 7, 2015
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2015 American Mathematical Society

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