Hopf-Galois extensions were introduced by Chase and Sweedler [CS69] in 1969, motivated by the problem of formulating an analogue of Galois theory for inseparable extensions. Their approach shed a new light on separable extensions. Later in 1987, the concept of Hopf-Galois theory was further developed by Greither and Pareigis [GP87]. So, as a problem in the theory of groups, they explained the problem of finding all Hopf-Galois structures on a finite separable extension of fields. After that, many results on Hopf-Galois structures were obtained by N. Byott, T. Crespo, S. Carnahan, L. Childs, and T. Kohl. In this thesis, we consider Hopf-Galois structures on Galois extensions of squarefree degree n. We first determine the number of isomorphism classes of groups G of order n whose centre and commutator subgroup have given orders, and we describe Aut(G) for each such G. By investigating regular cyclic subgroups in Hol(G), we enumerate the Hopf-Galois structures of type G on a cyclic extension of fields L/K of degree n. We then determine the total number of Hopf-Galois structures on L/K. Finally, we examine Hopf-Galois structures on a Galois extension L/K with arbitrary Galois group Gamma of order n, and give a formula for the number of Hopf-Galois structures on L/K of a given type G.

Cette thèse concerne quelques problèmes liés à la répartition des entiers sans facteur carré dansles progressions arithmétiques. Ces problèmes s’expriment en termes de majorations du terme d’erreurassocié à cette répartition.Les premier, deuxième et quatrième chapitres sont concentrés sur l’étude statistique des termesd’erreur quand on fait varier la progression arithmétique modulo q. En particulier on obtient une formuleasymptotique pour la variance et des majorations non triviales pour les moments d’ordre supérieur. Onfait appel à plusieurs techniques de théorie analytique des nombres comme les méthodes de crible et lessommes d’exponentielles, notamment une majoration récente pour les sommes d’exponentielles courtesdue à Bourgain dans le deuxième chapitre.Dans le troisième chapitre on s’intéresse à estimer le terme d’erreur pour une progression fixée. Onaméliore un résultat de Hooley de 1975 dans deux directions différentes. On utilise ici des majorationsrécentes de sommes d’exponentielles courtes de Bourgain-Garaev et de sommes d’exponentielles torduespar la fonction de Möbius dues à Bourgain et Fouvry-Kowalski-Michel. / This thesis concerns a few problems linked with the distribution of squarefree integers in arithmeticprogressions. Such problems are usually phrased in terms of upper bounds for the error term relatedto this distribution.The first, second and fourth chapter focus on the satistical study of the error terms as the progres-sions varies modulo q. In particular we obtain an asymptotic formula for the variance and non-trivialupper bounds for the higher moments. We make use of many technics from analytic number theorysuch as sieve methods and exponential sums. In particular, in the second chapter we make use of arecent upper bound for short exponential sums by Bourgain.In the third chapter we give estimates for the error term for a fixed arithmetic progression. Weimprove on a result of Hooley from 1975 in two different directions. Here we use recent upper boundsfor short exponential sums by Bourgain-Garaev and exponential sums twisted by the Möbius functionby Bourgain et Fouvry-Kowalski-Michel.