Let L/K be a 2-birational CM-extension of a totally real 2-rational number field. We characterize in terms of tame ramification totally real 2-extensions K’/K such that the compositum L’=LK’ is still 2-birational. In case the 2-extension K’/K is linearly disjoint from the cyclotomic ℤ₂-extension , we prove that K’/K is at most quadratic. Furthermore, we construct infinite towers of such 2-extensions.
We study the ramification properties of the extensions under the hypothesis that is odd and if than either or ( and are the exponents with which divides and ). In particular we determine the higher ramification groups of the completed extensions and the Artin conductors of the characters of their Galois group. As an application, we give formulas for the -adique valuation of the discriminant of the studied global extensions with .
Nous appliquons à la notion d’extension (cyclique de degré ) à ramification minimale, les techniques de “ réflexion ” qui permettent une caractérisation très simple de ces extensions à l’aide d’un corps gouvernant.
We begin by giving a criterion for a number field with 2-class group of rank 2 to have a metacyclic Hilbert 2-class field tower, and then we will determine all real quadratic number fields that have a metacyclic nonabelian Hilbert -class field tower.
A representation field for a non-maximal order in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders containing a copy of . Not every non-maximal order has a representation field. In this work we prove that every commutative order has a representation field and give a formula for it. The main result is proved for central simple algebras over arbitrary global fields.
Let be a set of binary quadratic forms of the same discriminant, a set of arithmetical progressions and a positive integer. We investigate the representability of prime powers lying in some progression from by some form from .
Nous introduisons les notions de nombres et d’idéaux infinitésimaux attachés à un corps de nombres algébriques relativement à un nombre premier donné , et nous interprétons le groupe de Galois de la -extension abélienne -ramifiée maximale de comme quotient du tensorisé du groupe des idéaux étrangers à par le sous-module engendré par les idéaux principaux-infinitésimaux. Nous en déduisons diverses conséquences sur l’arithmétique des groupes , en montrant en particulier qu’ils donnent...