On the Galois correspondence ratio for Hopf-Galois extensions arising from nilpotent \mathbbF_p-algebras

2023/06/15 by Lindsay N. Childs, Childs, Lindsay N.
Mathematics · #12F10 #16N40 #16T05 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2306.09163

openalex publication_date 2023/06/15 · openalex created_date 2023/06/17 · openalex updated_date 2026/07/28

Abstract

For a Hopf-Galois structure on a Galois extension L/K of fields that arises from a finite nilpotent \mathbbF_p-algebra A, we look at the Galois correspondence ratio, which measures the failure of surjectivity of the Galois correspondence for the Hopf-Galois structure on L/K. Using methods of elementary linear algebra, we observe that the number of subgroups of the adjoint group of A is equal to the number of subgroups of the additive group of N. Then we count left ideals of A and thereby determine the GCR for all nilpotent \mathbbF_p-algebras of dimension 4, and also show that for a set of \mathbbF_p-algebras of arbitrary dimension n and exponent e, the GCR approaches 0 for large p, n or e.

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