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Truman, PJ (2020) Hopf-Galois module structure of tamely ramified radical extensions of prime degree. Journal of Pure and Applied Algebra, 224 (5). ISSN 0022-4049
HGMS of tamely ramified radical extensions of prime degree RESUBMISSION.pdf - Accepted Version
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Abstract
Let K be a number field and let L/K be a tamely ramified radical extension of prime degree p. If K contains a primitive p th root of unity then L/K is a cyclic Kummer extension; in this case the group algebra K[G] (with G = Gal(L/K)) gives the unique Hopf-Galois structure on L/K, the ring of algebraic integers OL is locally free over OK[G] by Noether’s theorem, and G´omez Ayala has determined a criterion for OL to be a free OK[G]-module. If K does not contain a primitive p th root of unity then L/K is a separable, but non-normal, extension, which again admits a unique Hopf-Galois structure. Under the assumption that p is unramified in K, we show that OL is locally free over its associated order in this Hopf-Galois structure and determine a criterion for it to be free. We find that the conditions that appear in this criterion are identical to those appearing in G´omez Ayala’s criterion for the normal case.
Item Type: | Article |
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Additional Information: | The final version of this accepted manuscript will be available online at https://doi.org/10.1016/j.jpaa.2019.106231 |
Uncontrolled Keywords: | Hopf-Galois structure, Hopf-Galois module theory, Galois module structure, Associated order |
Subjects: | R Medicine > R Medicine (General) R Medicine > RA Public aspects of medicine |
Divisions: | Faculty of Natural Sciences > School of Chemical and Physical Sciences |
Depositing User: | Symplectic |
Date Deposited: | 12 Sep 2019 08:05 |
Last Modified: | 12 Sep 2020 01:30 |
URI: | https://eprints.keele.ac.uk/id/eprint/6810 |