The ring of integers of Hopf-Galois degree p extensions of p-adic fields with dihedral normal closure

2022/05/26 by Gil-Muñoz, Daniel
#11S15 #12F10 #16T05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2205.13517

Abstract

For an odd prime number p, we consider degree p extensions L/K of p-adic fields with normal closure \widetildeL such that the Galois group of \widetildeL/K is the dihedral group of order 2p. We shall prove a complete characterization of the freeness of the ring of integers O_L over its associated order \mathfrakA_L/K in the unique Hopf-Galois structure on L/K, which is analogous to the one already known for cyclic degree p extensions of p-adic fields. We shall derive positive and negative results on criteria for the freeness of O_L as \mathfrakA_L/K-module.

Related

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• Hopf Galois module structure of dihedral degree 2p extensions of ℚ_p
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• Hopf-Galois Structures on Non-Normal Extensions of Degree Related to\n Sophie Germain Primes
• Extensions of degree p^l of a p-adic field