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We study the Hopf-Galois module structure of algebraic integers in some finite extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is a finite unramified extension of $ p $-adic fields which is Hopf-Galois for some Hopf algebra $ H $ then the ring of algebraic integers $ \OL $ is a free module of rank one over the associated order $ \AH $. If $ H $ is a commutative Hopf algebra, we show that this conclusion remains valid in finite ramified extensions of $ p $-adic fields if $ p $ does not divide the degree of the extension. We prove analogous results for finite abelian Galois extensions of number fields, in particular showing that if $ L/K $ is a finite abelian domestic extension which is Hopf-Galois for some commutative Hopf algebra $ H $ then $ \OL $ is locally free over $ \AH $. We study in greater detail tamely ramified Galois extensions of number fields with Galois group isomorphic to $ C_{p} \times C_{p} $, where $ p $ is a prime number. Byott has enumerated and described all the Hopf-Galois structures admitted by such an extension. We apply the results above to show that $ \OL $ is locally free over $ \AH $ in all of the Hopf-Galois structures, and derive necessary and sufficient conditions for $ \OL $ to be globally free over $ \AH $ in each of the Hopf-Galois structures. In the case $ p = 2 $ we consider the implications of taking $ K = \Q $. In the case that $ p $ is an odd prime we compare the structure of $ \OL $ as a module over $ \AH $ in the various Hopf-Galois structures.

Let K be a number field with ring of integers O_K and let G be a finite group.By a result of E. Noether, the ring of integers of a tame Galois extension of K with Galois group G is a locally free O_K[G]-module of rank 1.Thus, to any tame Galois extension L/K with Galois group G we can associate a class [O_L] in the locally free class group Cl(O_K[G]). The set of all classes in Cl(O_K[G]) which can be obtained in this way is called the set of realizable classes and is denoted by R(O_K[G]).In this dissertation we study different problems related to R(O_K[G]).The first part focuses on the following question: is R(O_K[G]) a subgroup of Cl(O_K[G])? When the group G is abelian, L. McCulloh proved that R(O_K[G]) coincides with the so-called Stickelberger subgroup St(O_K[G]) of Cl(O_K[G]). In Chapter 2, we give a detailed presentation of unpublished work by L. McCulloh that extends the definition of St(O_K[G]) to the non-abelian case and shows that R(O_K[G]) is contained in St(O_K[G]) (the opposite inclusion is still not known in the non-abelian case).Then, just using its definition and Stickelberger's classical theorem, we prove in Chapter 3 that St(O_K[G]) is trivial if K=Q and G is either cyclic of order p or dihedral of order 2p, where p is an odd prime number. This, together with McCulloh's results, allows us to have a new proof of the triviality of R(O_K[G]) in the cases just considered.The main original results are contained in the second part of this thesis. In Chapter 4, we prove that St(O_K[G]) has good functorial behavior under restriction of the base field. This has the interesting consequence that, if N/L is a tame Galois extension with Galois group G, and St(O_K[G]) is known to be trivial for some subfield K of L, then O_N is stably free as an O_K[G]-module.In the last chapter, we prove an equidistribution result for Galois module classes amongst tame Galois extensions of K with Galois group G in which a given prime p of K is totally split.

Soient K un corps de nombres d'anneau des entiers O_K et G un groupe fini. Grâce à un résultat de E. Noether, l'anneau des entiers d'une extension galoisienne de K modérément ramifiée, de groupe de Galois G, est un O_K[G]-module localement libre de rang 1. Donc, à chaque extension galoisienne L/K modérément ramifiée, de groupe de Galois G, on peut associer une classe [O_L] dans le groupe des classes des modules localement libres Cl(O_K[G]). L'ensemble des classes de Cl(O_K[G]) qui peuvent être obtenues de cette façon est appelé ensemble des classes réalisables et on le note R(O_K[G]).Dans cette thèse, on étudie différents problèmes liés à R(O_K[G]). Dans la première partie, nous nous focalisons sur la question suivante: R(O_K[G]) est-il un sous-groupe de Cl(O_K[G])? Si G est abélien, L. McCulloh a prouvé que R(O_K[G]) coïncide avec le soi-disant sous-groupe de Stickelberger St(O_K[G]) dans Cl(O_K[G]). Dans le Chapitre 2, nous donnons une présentation détaillée d'un travail non publié de L. McCulloh qui étend la définition de St(O_K[G]) au cas non-abélien et montre que R(O_K[G]) est inclus dans St(O_K[G]) (l'inclusion opposée n'est pas encore connue dans le cas non-abélien). Puis, en utilisant sa définition et le Théorème de Stickelberger classique, nous montrons dans le Chapitre 3 que St(O_K[G]) est trivial si K=Q et G est soit un groupe cyclique d'ordre p soit un groupe diédral d'ordre 2p, avec p premier impair. Ceci, lié aux résultats de McCulloh, nous donne une nouvelle preuve de la trivialité de R(O_K[G]) dans les cas considérés.Les résultats originaux les plus importants sont contenus dans la deuxième partie de cette thèse. Dans le Chapitre 4 nous montrons la fonctorialité de St(O_K[G]) par rapport au changement du corps de base. Ceci implique que si N/L est une extension galoisienne modérément ramifiée, de groupe de Galois G, et St(O_K[G]) est connu être trivial pour un certain sous-corps K de L, alors O_N est un O_K[G]-module stablement libre.Dans le dernier chapitre, nous montrons un résultat concernant la distribution des classes réalisables parmi les extensions galoisiennes de K modérément ramifiées, de groupe de Galois G, dans lesquelles un idéal premier de K donné est totalement décomposé. / Let K be a number field with ring of integers O_K and let G be a finite group.By a result of E. Noether, the ring of integers of a tame Galois extension of K with Galois group G is a locally free O_K[G]-module of rank 1.Thus, to any tame Galois extension L/K with Galois group G we can associate a class [O_L] in the locally free class group Cl(O_K[G]). The set of all classes in Cl(O_K[G]) which can be obtained in this way is called the set of realizable classes and is denoted by R(O_K[G]).In this dissertation we study different problems related to R(O_K[G]).The first part focuses on the following question: is R(O_K[G]) a subgroup of Cl(O_K[G])? When the group G is abelian, L. McCulloh proved that R(O_K[G]) coincides with the so-called Stickelberger subgroup St(O_K[G]) of Cl(O_K[G]). In Chapter 2, we give a detailed presentation of unpublished work by L. McCulloh that extends the definition of St(O_K[G]) to the non-abelian case and shows that R(O_K[G]) is contained in St(O_K[G]) (the opposite inclusion is still not known in the non-abelian case).Then, just using its definition and Stickelberger's classical theorem, we prove in Chapter 3 that St(O_K[G]) is trivial if K=Q and G is either cyclic of order p or dihedral of order 2p, where p is an odd prime number. This, together with McCulloh's results, allows us to have a new proof of the triviality of R(O_K[G]) in the cases just considered.The main original results are contained in the second part of this thesis. In Chapter 4, we prove that St(O_K[G]) has good functorial behavior under restriction of the base field. This has the interesting consequence that, if N/L is a tame Galois extension with Galois group G, and St(O_K[G]) is known to be trivial for some subfield K of L, then O_N is stably free as an O_K[G]-module.In the last chapter, we prove an equidistribution result for Galois module classes amongst tame Galois extensions of K with Galois group G in which a given prime p of K is totally split.

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