Local Galois Module Structure in Characteristic p
Marklove, Maria Louise
Thesis or dissertation
University of Exeter
For a finite, totally ramified Galois extension L/K (of prime degree p) of local fields of characterstic p, we investigate the embedding dimension of the associated order, and the minimal number of generators over the associated order, for an arbitrary fractional ideal in L. This is intricately linked to the continued fraction expansion of s/p, where s is the ramification number of the extension. This investigation can be thought of as a generalisation of 'Local Module Structure in Positive Characteristic' (de Smit & Thomas, Arch. Math 2007) - which was concerned with the rings of integers only - and also as a specific, worked example of the more general 'Scaffolds and Generalized Integral Galois Module Structure' (Byott & Elder, arXiv:1308.2088[math.NT], 2013) - which deals with degree pk extensions, for some k, which admit a Galois scaffold. We also obtain necessary and sufficient conditions for the freeness of these ideals over their associated orders. We show these conditions agree with the analogous conditions in the characteristic 0 case, as described in 'Sur les ideaux d'une extension cyclique de degre premier d'un corps local' (Ferton, C.R. Acad. Sc. Paris, 1973).
PhD in Mathematics