Skew braces and the Galois correspondence for Hopf Galois structures

摘要

LetL/Kbe a Galois extension of fields with Galois group Γ, and supposeL/Kis also anH-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 ofHto intermediate subfields ofL/K, given by the Fundamental Theorem of Hopf Galois Theory. SupposeL?KH=LNwhereN?(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 fieldsEbetweenKandLthat correspond toK-subHopf algebras ofHand a set of sub-skew left braces ofGthat we call the °-stable subgroups of(G,?). Counting these subgroups and comparing that number with the number of subgroups ofΓ?(G,°)describes how far the Galois correspondence for theH-Hopf Galois structure is from being surjective. The method is illustrated by a variety of examples.