The Hopf Galois property in subfield lattices
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Let K/k be a finite separable extension, n its degree and (K) over tilde /k its Galois closure. For n <= 5, Greither and Pareigis show that all Hopf Galois extensions are either Galois or almost classically Galois and they determine the Hopf Galois character of K/ k according to the Galois group (or the degree) of (K) over tilde /k. In this paper we study the case n = 6, and intermediate extensions F/ k such that K subset of F subset of (K) over tilde, for degrees n = 4, 5, 6. We present an example of a non almost classically Galois Hopf Galois extension of (sic) of the smallest possible degree and new examples of Hopf Galois extensions. In the last section we prove a transitivity property of the Hopf Galois condition.
CitacióCrespo, T., Rio, A., Vela, M. The Hopf Galois property in subfield lattices. "Communications in algebra", 01 Gener 2016, vol. 44, núm. 1, p. 336-353.