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

For an odd prime number p, we consider degree p extensions L/K of p-adic fields with normal closure L˜ such that the Galois group of L˜/K is the dihedral group of order 2p. We shall prove a complete characterization of the freeness of the ring of integers OL over its associated order AL/K in the uni...

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Bibliographic Details
Published inJournal of number theory Vol. 245; pp. 65 - 118
Main Author Gil-Muñoz, Daniel
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.04.2023
Subjects
Associated order
Ramification jump
Ring of integers
Associated order
16T05
Ring of integers
11S15
Ramification jump
12F10
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Summary:For an odd prime number p, we consider degree p extensions L/K of p-adic fields with normal closure L˜ such that the Galois group of L˜/K is the dihedral group of order 2p. We shall prove a complete characterization of the freeness of the ring of integers OL over its associated order AL/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 OL as AL/K-module.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2022.10.002