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 $p^{\prime }(x)$ is the derivative of ${\pi }_L$'s minimal polynomial over $K$, then the relative different $\euD_{L/K}$ is generated by $p^{\prime }({\pi }_L)\in L$. Let $v_L$ be the normalized valuation normalized with $v_L(L)=\mathbb{Z}$. We show that any element $\rho \in L$ with $v_L(\rho )\equiv -v_L(p^{\prime }({\pi }_L))-1\mathrm{mod}[L:K]$ generates a normal basis, $K[Gal(L/K)]\cdot \rho =L$. This criterion is tight: Given any integer $i$ such that $i\not\equiv -v_L(p^{\prime }({\pi }_L))-1\mathrm{mod}[L:K]$, there is a ${\rho }_i\in L$ with $v_L({\rho }_i)=i$ such that $K[Gal(L/K)]\cdot {\rho }_i⊊L$.