Hopf-Galois module structure of some non-normal extensions

Prestidge, George Thomas

Abstract

We use Hopf-Galois theory to study the structure of rings of algebraic integers in some non-normal extensions of number fields which are tamely ramified, generalising results of Del Corso and Rossi for tamely ramified Kummer extensions.

Firstly we study tamely ramified non-normal extensions of number fields of the form L = K( p √ a1, ..., p √ ar) for some prime number p and a1, ..., ar ∈ OK. We show that extensions of this form admit a unique almost classical Hopf-Galois structure and that if r = 2 then this is the only Hopf-Galois structure on the extension. We then obtain explicit OK,p-bases of OL,p for each prime ideal p of OK. Using these, we show that OL is locally free over its associated order in the unique almost classical Hopf-Galois structure on the extension. To obtain criteria for OL to be free over this associated order we use an id`elic description of the locally free class group of the maximal order.
Secondly we conduct an analogous study of tamely ramified non-normal extensions of number fields of the form L = K( m √ a) for some odd square-free number m = p1...pr and a ∈ OK. Once again, we find that extensions of this form admit a unique almost classical Hopf-Galois structure. Once again we show that if r = 2 then this is the only Hopf-Galois structure on the extension. We again use explicit OK,p-bases of OL,p for each prime ideal p of OK to show that OL is locally free over its associated order in the almost classical Hopf-Galois structure on the extension. Once again, to obtain criteria for OL to be free over this associated order we use an id`elic description of the locally free class group of the maximal order.

In both cases, the criteria we obtain are identical to those obtained by Del Corso and Rossi in the Galois case.

Citation

Prestidge, G. T. (2024). Hopf-Galois module structure of some non-normal extensions. (Thesis). Keele University. Retrieved from https://keele-repository.worktribe.com/output/850279

• Thesis Type: Thesis
• Deposit Date: Jun 14, 2024
• Publicly Available Date: Jun 20, 2024
• Public URL: https://keele-repository.worktribe.com/output/850279
• Award Date: 2024-06