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Soient des nombres premiers distincts , et . On peut approcher le -rang du groupe de classes des corps en étudiant celui du corps pour un entier . Dans cet article, on traite le cas où ou . Comme application, on déduit que le rang du -groupe de classes de est au moins égal à deux (on savait déjà grâce à un résultat de Fröhlich que le groupe de classes de est toujours d’ordre pair). On en déduit également la liste de tous les corps multiquadratiques ayant un -groupe de classes...
Let be a number field with ring of integers . For a fixed prime number and the étale wild kernels are defined as kernels of certain localization maps on the -fold twist of the -adic étale cohomology groups of . These groups are finite and coincide for with the -part of the classical wild kernel . They play a role similar to the -part of the -class group of . For class groups, Galois co-descent in a cyclic extension is described by the ambiguous class formula given by genus theory....
Let be an odd prime and a cyclic -extension of number fields. We give a lower bound for the order of the kernel and cokernel of the natural extension map between the even étale -groups of the ring of -integers of , where is a finite set of primes containing those which are -adic.
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