Hopf Galois structures on Kummer extensions of prime power degree (Q554236)

Hopf Galois structures on separable field extensions \(L/K\) were studied by \textit{B. Pareigis} and the reviewer [J. Algebra 106, 239--258 (1987; Zbl 0615.12026)]. The description of the possible Hopf algebras \(H\) together with their action on \(L\) comes down, via descent and Galois theory, to a group theoretical problem. This was made much more amenable by \textit{N. P. Byott} [Commun. Algebra 24, No. 10, 3217--3228 (1996; Zbl 0878.12001)]. \textit{T. Kohl} [J. Algebra 207, No. 2, 525--546 (1998; Zbl 0953.12003)] showed that if \(L/K\) is Galois (in the classical sense) with cyclic Galois group of odd prime power order \(p^n\), then there exist exactly \(p^{n-1}\) Hopf Galois structures on \(L/K\). Since this result used the group-theoretical approach, it did not reveal the explicit structure of the Hopf algebra \(H\) and its action on \(L\). One should note that the structure is clear for \(n=1\) (there is only the classical Hopf structure), and for \(n=2\) one can extract it from work of the reviewer [Math. Z. 210, No. 1, 37--68 (1992; Zbl 0737.11038)]. For \(n>2\) nothing was known. The article under review remedies this by determining \(H\) explicitly as a subalgebra of a certain group algebra \(L[M]\) over \(L\). The main effort is required for finding a set of algebra generators. The comultiplication and the action on \(L\) are inherited from \(L[M]\). The results are very technical (the reviewer refrains from even stating them here) and the proofs involve long calculations, but this is novel and of interest to people working in Hopf Galois theory.