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

##### View/Open

Accepted author manuscript (328.5Kb)
##### Date

2015##### Author

Johnston, Henri

##### Date issued

2015

##### Journal

Proceedings of the American Mathematical Society

##### Type

Article

##### Language

en

##### Publisher

American Mathematical Society

##### 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.

##### Description

First published in Proceedings of the American Mathematical Society in Vol 143 (2015), published by the American Mathematical Society

##### Citation

Vol. 143, pp. 5059-5071

##### EISSN

1088-6826

##### ISSN

0002-9939