Hopf-Galois Structures on Non-Normal Extensions of Degree Related to Sophie Germain Primes
| Autor: | Byott, Nigel P., Martin-Lyons, Isabel |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions $L/K$ of squarefree degree $n$. If $E/K$ is the normal closure of $L/K$ then $G=\mathrm{Gal}(E/K)$ can be viewed as a permutation group of degree $n$. We show that $G$ has derived length at most $4$, but that many permutation groups of squarefree degree and of derived length $2$ cannot occur. We then investigate in detail the case where $n=pq$ where $q \geq 3$ and $p=2q+1$ are both prime. (Thus $q$ is a Sophie Germain prime and $p$ is a safeprime). We list the permutation groups $G$ which can arise, and we enumerate the Hopf-Galois structures for each $G$. There are six such $G$ for which the corresponding field extensions $L/K$ admit Hopf-Galois structures of both possible types. Comment: 26 pages. Proposition 4.12 and Table 5 corrected. Theorem 3.5 removed. Typos fixed and some improvements made to wording. Accepted for Journal of Pure and Applied Algebra |
| Databáze: | arXiv |
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