2021/02/10 by Nigel P. Byott, Byott, Nigel P., Isabel Martin-Lyons +1 · 1 citation
Mathematics · #12F10 #16T05 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2102.05759
openalex publication_date 2021/02/10 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We consider Hopf-Galois structures on separable (but not necessarily normal)\nfield extensions L/K of squarefree degree n. If E/K is the normal closure\nof L/K then G=\Gal(E/K) can be viewed as a permutation group of\ndegree n. We show that G has derived length at most 4, but that many\npermutation groups of squarefree degree and of derived length 2 cannot occur.\nWe then investigate in detail the case where n=pq where q \≥ 3 and\np=2q+1 are both prime. (Thus q is a Sophie Germain prime and p is a\nsafeprime). We list the permutation groups G which can arise, and we\nenumerate the Hopf-Galois structures for each G. There are six such G for\nwhich the corresponding field extensions L/K admit Hopf-Galois structures of\nboth possible types.\n