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A Hopf Galois structure on a finite field extension $L/K$ is a pair$(H,\mu)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ aHopf action. In this paper we present a program written in the computationalalgebra system Magma which gives all Hopf Galois structures on separable fieldextensions of degree up to eleven and several properties of those. Besides, weexhibit several results on Hopf Galois structures inspired by the programoutput. We prove that if $(H,\mu)$ is an almost classically Hopf Galoisstructure, then it is the unique Hopf Galois structure with underlying Hopfalgebra $H$, up to isomorphism. For $p$ an odd prime, we prove that a separableextension of degree $p^2$ may have only one type of Hopf Galois structure anddetermine those of cyclic type; we determine as well the Hopf Galois structureson separable extensions of degree $2p$. We highlight the richness of theresults obtained for extensions of degree 8 by computing an explicit exampleand presenting some tables which summarizes these results.

Computation of Hopf Galois structures on low degree separable extensions and classification of those for degrees $p^2$ and $2p$