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We determine all Hopf-Galois structures on a Galois extension of ÿelds of degree pq, where p, q are primes with p ≡ 1 (mod q). There are 2q − 1, respectively 2 + p(2q − 3), Hopf-Galois structures when the extension is cyclic, respectively nonabelian. Explicit generators are given for the groups of permutations corresponding to these Hopf-Galois structures.

Let k be a number field and O k its ring of integers. Let G be a finite group, N=k a Galois extension with Galois group isomorphic to G, and O N the ring of integers of N. Let M be a maximal O k -order in the semi-simple algebra k½G containing O k ½G, and ClðMÞ its locally free classgroup. When N=k is tame (i.e., at most tamely ramified), extension of scalars allows us to assign toin ClðMÞ. We define the set RðMÞ of realizable classes to be the set of classes c A ClðMÞ such that there exists a Galois extension N=k which is tame, with Galois group isomorphic to G, and for which ½M n O k ½G O N ¼ c. In the present article, we prove, by means of a fairly explicit description, that RðMÞ is a subgroup of ClðMÞ when G ¼ V z r C, where V is an F 2 -vector space of dimension r f 2, C a cyclic group of order 2 r À 1, and r a faithful representation of C in V ; an example is the alternating group A 4 . In the proof, we use some properties of the binary Hamming code and solve an embedding problem connected with Steinitz classes. In addition, we determine the set of Steinitz classes of tame Galois extensions of k, with the above group as Galois group, and prove that it is a subgroup of the classgroup of k.

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