Journal de Théorie des Nombres

A classification of the extensions of degree p2 over Qp whose normal closure is a p-extension par Luca CAPUTO Résumé. Soit k une extension finie de Qp et soit Ek l’ensemble des extensions de degre ́ p2 sur k dont la clôture normale est une p-extension. Pour chaque discriminant fixé, nous calculons le nombre des éléments de EQp qui ont un tel discriminant et nous donnons les discriminants et les groupes de Galois (avec leur fil-trations des groupes de ramification) de leurs clôtures normales. Nous montrons aussi que l’on peut generalizer cette méthode pour obtenir une classification des extensions qui appartiennent a ̀ Ek. Abstract. Let k be a finite extension of Qp and Ek be the set of the extensions of degree p2 over k whose normal closure is a p-extension. For a fixed discriminant, we show how many exten-sions there are in EQp with such discriminant and we give the dis-criminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalized to get a classification of the extensions in Ek. 1. Notation, preliminaries and results. Throughout this paper, p is an odd prime and k will be a fixed p-adic field of degree d over Qp which does not contain any primitive p-th root of unity. If E is a p-adic field and L|E is a finite extension, then we say that L|E is a p-extension if it is Galois and its degree is a power of p. The aim of the present paper is to give a classification of the extensions of degree p2 over Qp whose normal closure is a p-extension. This classification is based on the discriminant of the extension and on the Galois group and the discriminant of its normal closure. Let Ek be the set of the extensions of degree p2 over k whose normal closure is a p-extension. Then for every L ∈ Ek, there exists a cyclic extension K|k of degree p, K ⊆ L and L|K is cyclic (of degree p). Furthermore, the converse is true: if K|k is a cyclic extension of degree p, then every cyclic extension L of degree p over K is an extension of degree p2 over k whose normal closure is a p-extension (see Prop. 2.1). Therefore, if K|k is a cyclic extension of degree p, we can ha l-