Sur quelques aspects des extensions à ramification restreinte

Abstract : Let p be a prime number, let K|k be a Galois extension of number fields and let S be a finite set of primes of K. We suppose that the degree of K|k is finite and coprime to p. We denote by G(K,S) the Galois group of the pro-p maximal extension of K unramified outside S. We focus on this thesis on two differents aspects of this pro-p group. We are first interested in the tame case : we suppose that S does not contain any place above p. The works of Labute, Minac and Schmidt about mild pro-p groups brought the first examples of groups G(K,S) of cohomological dimension less two. Using a corollary of their criterium, we compute some examples with PARI/GP and we observe a propagation phenomenum : if we take k=Q and if we suppose that G(Q,S) is mild, a large part of the pro-p groups G(K,S) with K imaginary quadratic are mild too. We then associate two oriented graphs to G(K,S) and we show a theoretical criterium proving mildness of some imaginary quadratic fields. We then consider the wild case where all the places dividing p belong to S. The Galois group Gal(K|k) acts on G(K,S). The action of Gal(K|k) is trivial on some quotients of G(K,S) ; we denote by G the maximal one and by H the corresponding closed subgroup of G(K,S). Maire has studied the Z_p[[G]]-freeness of the module H^{ab}. We extend his results considering the phi-component of H^{ab} under the action of Gal(K|k). In a favourable context and under Leopoldt's conjecture, we show a necessary and sufficient condition for the freeness of the phi-components. This condition is connected to p-rational fields by class field theory. We present experiments with PARI/GP in some families of cubic cyclic, dihedral and quartic cyclic extensions of Q which support the following conjecture from Gras : every number field is p-rational for sufficiently large p.
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Marine Rougnant. Sur quelques aspects des extensions à ramification restreinte. Théorie des nombres [math.NT]. Université de Bourgogne Franche-Comté, 2018. Français. ⟨tel-01803456⟩



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