The valuation criterion for normal basis generators

TL;DR

This paper characterizes when a valuation-based criterion for normal basis generators holds in local field extensions, linking it to ramification, module structure, and characteristic, with explicit conditions for abelian extensions.

Contribution

It provides a complete characterization of the valuation criterion for normal basis generators in local fields, including explicit conditions and the case distinctions based on characteristic.

Findings

VC(L/K) holds iff tamely ramified part is trivial and certain modules contain units.

When char(K)>0, VC(L/K) holds for all extensions.

In char(K)=0, the paper classifies abelian extensions satisfying VC(L/K).

Abstract

If $L/K$ is a finite Galois extension of local fields, we say that the valuation criterion $VC(L/K)$ holds if there is an integer $d$ such that every element $x \in L$ with valuation $d$ generates a normal basis for $L/K$. Answering a question of Byott and Elder, we first prove that $VC(L/K)$ holds if and only if the tamely ramified part of the extension $L/K$ is trivial and every non-zero $K[G]$-submodule of $L$ contains a unit. Moreover, the integer $d$ can take one value modulo $[L:K]$ only, namely $-d_{L/K}-1$, where $d_{L/K}$ is the valuation of the different of $L/K$. When $K$ has positive characteristic, we thus recover a recent result of Elder and Thomas, proving that $VC(L/K)$ is valid for all extensions $L/K$ in this context. When $\char{\;K}=0$, we identify all abelian extensions $L/K$ for which $VC(L/K)$ is true, using algebraic arguments. These extensions are determined by the behaviour of their cyclic Kummer subextensions.