A valuation criterion for normal basis generators in equal positive characteristic

We answer a recent conjecture of [N.P. Byott, G.G. Elder, A valuation criterion for normal bases in elementary abelian extensions, Bull. London Math. Soc. 39 (5) (2007) 705–708] in a more general setting. Precisely, let L/K be a finite abelian p-extension of local fields of characteristic p>0 that is totally ramified. Let b denote the largest ramification break in the lower numbering. We prove that any element x L whose valuation over L is equal to b modulo [L:K] generates a normal basis of L/K. The arguments will develop certain properties of ramification groups and jumps, as well as the algebraic structure of certain group algebras.