Let N/F be an odd degree Galois extension of number fields with Galois group G and rings of integers ON and OF = O respectively. Let A be the unique fractional ON -ideal with square equal to the inverse different of N/F . Erez has shown that A is a locally free O[G]-module if and only if N/F is a so called weakly ramified extension. There have been a number of results regarding the freeness of A as a Z[G]-module, however this question remains open. In this paper we prove that A is free as a Z[G]-module assuming that N/F is weakly ramified and under the hypothesis that for every prime ℘ of O which ramifies wildly in N/F , the decomposition group is abelian, the ramification group is cyclic and ℘ is unramified in F/Q.We make crucial use of a construction due to the first named author which uses Dwork's exponential power series to describe self-dual integral normal bases in Lubin-Tate extensions of local fields. This yields a new and striking relationship between the local norm-resolvent and Galois Gauss sum involved. Our results generalise work of the second named author concerning the case of base field Q.

Let K be a finite extension of Q p , let L/K be a finite abelian Galois extension of odd degree and let O L be the valuation ring of L. We define A L/K to be the unique fractional O L -ideal with square equal to the inverse different of L/K . For p an odd prime and L/Q p contained in certain cyclotomic extensions, Erez has described integral normal bases for A L/Qp that are self-dual with respect to the trace form. Assuming K /Q p to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions.

Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that T r F/E (g(x), h(x)) = δ g,h for g, h ∈ Γ. Bayer-Fluckiger and Lenstra have shown that when char(E) = 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char(E) = 2, then F admits a self-dual normal basis if and only if the exponent of Γ is not divisible by 4. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let K be a finite extension of Q p , let L/K be a finite abelian Galois extension of odd degree and let O L be the valuation ring of L. We define A L/K to be the unique fractional O L -ideal with square equal to the inverse different of L/K. It is known that a self-dual integral normal basis exists for A L/K if and only if L/K is weakly ramified. Assuming p = 2, we construct such bases whenever they exist.

Recent work of Pickett has given a construction of self-dual normal bases for extensions of finite fields, whenever they exist. In this article we present these results in an explicit and constructive manner and apply them, through computer search, to identify the lowest complexity of self-dual normal bases for extensions of low degree. Comparisons to similar searches amongst normal bases show that the lowest complexity is often achieved from a self-dual normal basis

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.