Quelques théorèmes de base normale d'entiers

Abstract

Let Γ be a finite group, K a number field and N a tame Galois extension of K with group Γ. In the class group C(Z[Γ]) we study the order of the class of the ring of integers of N. We prove that this order divides [N:K] and that is 1 or 2 when Γ is metabelian or quaternionian, this depending on the Artin root numbers of the symplectic characters of Γ. We deduce new examples of extensions of Q with normal integral basis.