Calculs explicites en théorie d'Iwasawa

Abstract : In the first chapter of this thesis we explain Leopoldt's conjecture and some equivalent formulations. Then we give an algorithm that checks this conjecture for a given prime p and a number field. Next we assume that this conjecture is true, and we study the torsion part of the Galois group of the maximal abelian p-ramified p-extension of a given number field. We present a method to compute the invariant factors of this finite group. In the third chapter we give an interpretation of our numrical result by heuristics “à la” Cohen-Lenstra. In the fourth and last chapter, using our algorithm which computes this torsion submodule, we give new examples of numbers fields which satisfy Greenberg's conjecture.
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Firmin Varescon. Calculs explicites en théorie d'Iwasawa. Théorie des nombres [math.NT]. Université de Franche-Comté, 2014. Français. ⟨NNT : 2014BESA2019⟩. ⟨tel-01922312⟩



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