Hopf–galois Structures on Field Extensions With Simple Galois Groups

Hopf–Galois structures on Galois field extensions of degree pq

“…(This can be avoided in the special cases considered in [8,2]). It therefore makes sense to investigate simultaneously the Hopf-Galois structures on all Galois extensions L=K of a given degree n, at least in cases where the number of isomorphism types of group of order n is small.…”

“…Childs [6] has recently obtained results for the case where G is the holomorph of a cyclic group of odd prime-power order p e , showing in particular that there are precisely 2 + 5p + 4pq Hopf-Galois structures if e = 1 and q = (p − 1)=2 is also an odd prime. On the other hand, if G is a nonabelian simple group, then there are only two Hopf-Galois structures [2]. A detailed exposition of this theory up to 2000 can be found in [5].…”

Elementary abelian Hopf Galois structures and polynomial formal groups

“…For other results on counting Hopf Galois structures on Galois extensions of fields, see [5,Chapter 2] for 20th century work, and [3,4,6]. …”

Hopf–Galois structures on almost cyclic field extensions of 2-power degree

“…Here h → (h) h (1) ⊗ h (2) is the comultiplication map H → H ⊗ K H , and ε : H → K is the augmentation of H . Then L is said to be an H -Galois extension of K if the K-linear homomorphism j : L ⊗ K H → End K (L), given by j (x ⊗ h)(y) = x(h · y), is an isomorphism.…”

“…Greither-Pareigis theory Definition 2.1. Let L/K be a finite extension of fields, and let H be a K-Hopf algebra acting on L so that L becomes an H -module algebra: (1) · x)(h (2) · y); (h · 1) = ε(h)1 for x, y ∈ L and h ∈ H.…”