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

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 $K[{Gal}(L/K)]\cdot\rho_i\subsetneq L$.