A constructive theory for extensions of p-adic fields

The subject of this thesis in the study of nite extensions of p-adic fields, in different aspects. Via the study of the Galois module of p-th power classes L =(L )p of a general Galois extension L=K of degree p, it is possible to deduce and classify the extensions of degree p2 of a p-adic field. We exhibit formulae counting how many times a certain group appears as Galois group of the normal closure, generalizing previous results. In general degree we give a synthetic formula counting isomorphism classes of extensions of fixed degree. The formula is obtained via Krasner formula and a simple group-theoretic Lemma allowing to reduce the problem to counting cyclic extensions, which can be done easily via local class eld theory. When K is an unrami ed extension of Qp we study the problem of giving necessary and su cient conditions on the coe cients of an Eisenstein polynomial for it to have a prescribed group as Galois group of the splitting field. The techniques introduced allow to recover very easily Lbekkouri's result on cyclic extensions of degree p2, and to give a complete description of the Galois group, with its rami cation filtration, for splitting fields of Eisenstein polynomials of degree p2 which are a general p-extension. We then show how the same methods can be used to characterize Eisenstein polynomials defining a cyclic extension of degree p3. We then study Eisenstein polynomials in general, describing a family of special reduced polynomials which provide almost unique generators of totally rami ed extensions, and a reduction algorithm. The number of special polynomials generating a fixed extension L=K is always smaller than the number of conjugates of L over K, so that each Galois extension is generated by exactly one special polynomial. We give an algorithm to recover all special polynomials generating one extension, and a criterion that allows to detect when the extension generated by an Eisenstein polynomial is different from a fixed extension whose special generators are all given, the criterion does not only depend only on the usual distance on the set of Eisenstein polynomials defined by Krasner and others. An algorithm to construct the special polynomial generating an abelian class eld is given, provided a suitable description of a candidate norm subgroup of K x.