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

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 $(σ_1-1)^i(σ_2-1)^j,\ 0\le i,j\le p-1,$ on the valuation filtration; here $σ_1,σ_2$ 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.

💡 Research Summary

The paper investigates the structure of associated Galois modules for wildly ramified extensions of complete discrete valuation fields, focusing on the construction of explicit bases for these modules and the related orders. Let $K/k$ be a totally ramified, possibly wildly ramified extension with Galois group $G$. For each integer $i$ the authors define a filtration $C_i={f\in K