Critère effectif de puissancep-ième dans un corps de nombres galoisien Original Research Article

Lete≠0 be an integer of a Galois extensionK/image (of degreenand Galois groupG), and letpbe a prime number. Let (var epsilon′σ)σset membership, variantGbe a family of approximations (in image) of suitablepth rootsvar epsilonσof theσ(e),σset membership, variantG. Using a general criterion of the second author, we prove that there existsΔ(explicitly computable) such that if var epsilon′σ−var epsilonσ<(1/n) Δ, for allσset membership, variantG, and ∑σset membership, variantG var epsilon′σ−m<Δ, formset membership, variantimage,m≠0, theneis ap-power inK×. Some consequences are given, such as the fact that the method of “dévissages” of cyclotomic units in real abelian fields (for the determination of fundamental units and class numbers) is now valid and considerably easier.