A valuation criterion for normal basis generators in local fields of characteristic $p$

TL;DR

This paper establishes a valuation-based criterion for identifying normal basis generators in certain local fields of characteristic p, linking valuations to Galois module generation and demonstrating the criterion's optimality.

Contribution

It introduces a precise valuation criterion for normal basis generators in fully ramified Galois p-extensions of local fields of characteristic p, and proves its optimality.

Findings

The criterion precisely characterizes normal basis generators based on valuation.

The criterion is shown to be tight and optimal.

Counterexamples are provided for elements not satisfying the criterion.

Abstract

Let $K$ be a complete local field of characteristic $p$ with perfect residue field. Let $L/K$ be a finite, fully ramified, Galois $p$-extension. If $\pi_L\in L$ is a prime element, and $p'(x)$ is the derivative of $\pi_L$'s minimal polynomial over $K$, then the relative different $\euD_{L/K}$ is generated by $p'(\pi_L)\in L$. Let $v_L$ be the normalized valuation normalized with $v_L(L)=\mathbb{Z}$. We show that any element $\rho\in L$ with $v_L(\rho)\equiv -v_L(p'(\pi_L))-1\bmod[L:K]$ generates a normal basis, $K[{Gal}(L/K)]\cdot\rho=L$. This criterion is tight: Given any integer $i$ such that $i\not\equiv -v_L(p'(\pi_L))-1\bmod[L:K]$, there is a $\rho_i\in L$ with $v_L(\rho_i)=i$ such that $K[{Gal}(L/K)]\cdot\rho_i\subsetneq L$.