• Hopf algebras
• Hopf Galois
• Hopf Galois correspondence
• minimal Hopf Galois structure
• simple field extension
• uniqueness of Hopf Galois structure

We study Hopf Galois structures and the Hopf Galois correspondence following a path that will eventually lead us to the study of Hopf Galois structures in the extreme situation when the Hopf algebra has only two sub-Hopf algebras: the so-called minimal Hopf Galois structures. Hopf Galois extensions can be defined by extending some properties of classical Galois extensions. In particular, it turns out that a finite separable extension L/K is Galois with group G if and only if the K-vector space homomorphism j:L⊗KK[G]→EndK(L) given by j(l⊗σ):m→lσ(m) is a bijection. Roughly speaking, let H be a finite cocommutative Hopf algebra, then, to obtain an H-Hopf Galois extension we just need to replace K[G], which is a cocommutative Hopf algebra, with H.So Hopf Galois extensions can be thought of as a generalization of Galois extensions, although there are some differences between them. The most significant one is that a given extension can have many different Hopf Galois structures even when it is a Galois extension. A question naturally arises: how many are these structures, and what do they look like? To face these problems we discuss a theorem by Greither and Pareigis which shows that if L/K is a finite separable extension with normal closure E and Galois groups G=Gal(E/K) and G′=Gal(E/L), there is a bijection between the Hopf Galois structures of L/K and the regular subgroups of S(GmodG′) normalized by the subgroup of left translations λ(G). In this context, the isomorphism type of the subgroup associated to a Hopf Galois structure H is called the type of H. We also present a Hopf Galois version (proved by Sweedler and Chase) of the correspondence theorem of classical Galois theory which provides an inclusion-reversing injection between sub-Hopf algebras and intermediate fields. Here is another difference from the classical Galois theory: recent papers provide many examples of the failure of the surjectivity of this correspondence. However, Greither and Pareigis prove the surjectivity in Sweedler and Chase’stheorem for a special class of Hopf Galois separable extensions: the almost classical Galois extensions. We include this result together with a reformulation made by Crespo, Rio and Vela which is closer to the classical correspondence theorem. Having turned the problem of finding Hopf Galois structures into a group-theoretical one, one may ask how many Hopf-Galois structures there are on a Galois extension with a certain type of Galois group. There is plenty of literature on Hopf Galois structures on various classes of field extensions. For example, we present a work by Kohl in which he shows that, given an odd prime p, there are p^(n−1) Hopf Galois structures on a cyclic extension of degree p^n, all of cyclic type. In particular, a cyclic extension of prime degree admits a unique Hopf Galois structure. More generally, as far as uniqueness is concerned, we propose a result by Byott which characterizes the Galois extensions with a unique Hopf Galois structure: a Galois extension with Galois group G has a unique Hopf Galois structure if and only if |G| is a Burnside number, namely (|G|,φ(|G|)) = 1. We also prove a recent generalization by Byott and Martin-Lyons: a separable extension of degree a Burnside number is Hopf Galois if and only if it is almost classically Galois and in this case, it has a unique Hopf Galois structure. Finally, we discuss a recent result by Byott which shows that a nonabelian simple Galois extension admits exactly two Hopf Galois structures, and we study Ezome and Greither’s results on group theoretical methods to determine minimal Hopf Galois structures on finite separable extensions.