Sur la structure galoisienne relative de puissances de la différente et idéaux de stickelberger

Abstract

Let [Formula: see text] be a number field, [Formula: see text] its ring of integers, [Formula: see text] its classgroup and [Formula: see text] the class number of [Formula: see text]. Let [Formula: see text] be a finite group. Let [Formula: see text] be a maximal [Formula: see text]-order in the semi-simple algebra [Formula: see text] containing [Formula: see text], and [Formula: see text] its locally free classgroup. Let [Formula: see text] and [Formula: see text]. We define the set [Formula: see text] of Galois module classes realizable by the [Formula: see text]th power of the different to be the set of classes [Formula: see text] such that there exists a Galois extension [Formula: see text] with Galois group isomorphic to [Formula: see text] ([Formula: see text]-extension), which is tamely ramified, and for which the class of [Formula: see text] is equal to [Formula: see text], where we clarify that if [Formula: see text], where [Formula: see text], [Formula: see text] is the [Formula: see text]th root of the inverse different [Formula: see text] (respectively, the different [Formula: see text]) if [Formula: see text] (respectively, [Formula: see text]) when it exists. Let [Formula: see text] be a prime number and [Formula: see text] be a primitive [Formula: see text]th root of unity. In this article, we suppose that [Formula: see text] is cyclic of order [Formula: see text] and [Formula: see text] and [Formula: see text] are linearly disjoint. We prove, sometimes under an assumption on [Formula: see text], that [Formula: see text] is a subgroup of [Formula: see text], by an explicit description using a Stickelberger ideal. In addition, for each [Formula: see text], we determine the set of the Steinitz classes of [Formula: see text], [Formula: see text] runs through the tame [Formula: see text]-extensions of [Formula: see text], and prove that it is a subgroup of [Formula: see text], also sometimes under an hypothesis on [Formula: see text].