No abstract

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.

The first chapter provides basic p~erequisites.In

Explicit descriptions of local integral Galois module generators in certain extensions of p-adic fields due to Pickett have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields. In parallel, Pulita has generalised the theory of Dwork's power series to a set of power series with coefficients in Lubin-Tate extensions of Q p to establish a structure theorem for rank one solvable p-adic differential equations.In this paper we first generalise Pulita's power series using the theories of formal group exponentials and ramified Witt vectors. Using these results and Lubin-Tate theory, we then generalise Pickett's constructions in order to give an analytic representation of integral normal basis generators for the square root of the inverse different in all abelian totally, weakly and wildly ramified extensions of a p-adic field. Other applications are also exposed.