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

Abstract

Let L / K be a finite separable field extension, and let E be the normal closure of L / K . Let G = Gal ( E / K ) and G ′ = Gal ( E / L ) . We call L / K almost cyclic if G ′ has a normal cyclic complement in G. This includes the case that L / K is a cyclic Galois extension or a radical extension. We give a method for counting Hopf–Galois structures on an almost cyclic extension L / K . We then count the Hopf–Galois structures on an almost cyclic extension of degree 2 n , n ⩾ 3 , and determine how many of them are almost classical. This is analogous to a result of T. Kohl [T. Kohl, Classification of the Hopf–Galois structures on prime power radical extensions, J. Algebra 207 (1998) 525–546] which counts the Hopf–Galois structures on a radical extension of odd prime-power degree. In contrast to the odd prime-power degree case, however, we find that an almost cyclic extension L / K of 2-power degree has Hopf–Galois structures for which the Hopf algebra acting on L is not commutative.