Skew braces and the Galois correspondence for Hopf Galois structures

Let L/K be a Galois extension of fields with Galois group Γ, and suppose L/K is also an H-Hopf Galois extension. Using the recently uncovered connection between Hopf Galois structures and skew left braces, we introduce a method to quantify the failure of surjectivity of the Galois correspondence from subHopf algebras of H to intermediate subfields of L/K, given by the Fundamental Theorem of Hopf Galois Theory. Suppose L⊗KH=LN where N≅(G,⋆). Then there exists a skew left brace (G,⋆,∘) where (G,∘)≅Γ. We show that there is a bijective correspondence between the set of intermediate fields E between K and L that correspond to K-subHopf algebras of H and a set of sub-skew left braces of G that we call the ∘-stable subgroups of (G,⋆). Counting these subgroups and comparing that number with the number of subgroups of Γ≅(G,∘) describes how far the Galois correspondence for the H-Hopf Galois structure is from being surjective. The method is illustrated by a variety of examples.