Bases of associated Galois modules in general wildly ramified extensions and in elementary abelian extensions of degree $p^2$

TL;DR

This paper investigates the structure of Galois modules in wildly ramified extensions, providing explicit computations for elementary abelian extensions of degree p^2 and conditions for constructing effective bases.

Contribution

It introduces a method to construct bases of Galois modules in wildly ramified extensions, with explicit calculations for degree p^2 elementary abelian extensions.

Findings

Explicit computation of Galois action on valuation filtration for G=(Z/pZ)^2

Identification of conditions under which constructed elements form effective bases

Analysis of ramification jumps modulo p^2 for basis effectiveness

Abstract

For a wildly ramified extension $K/k$ of complete discrete valuation fields we study collections of elements of $k[G]$ (where $G=Gal(K/k)$) that fit well for constructing bases of various associated Galois modules and orders. In the case $G=(Z/pZ)^2$ (where $p$ is the characteristic of residue fields) we are able to compute the action of the elements $(\sigma_1-1)^i(\sigma_2-1)^j,\ 0\le i,j\le p-1,$ on the valuation filtration; here $\sigma_1,\sigma_2$ are generators of $G$. If the ramification jumps of $K/k$ are distinct modulo $p^2$ then these elements do yield "good enough" bases in question.