The Hasse norm principle for some non-Galois extensions of square-free degree
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913810934136832 |
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| author | Oki, Yasuhiro |
| author_facet | Oki, Yasuhiro |
| contents | In this paper, we study the Hasse norm principle for some non-Galois extensions of number fields. Our main theorem is that for any square-free composite number $d$ which is divisible by at least one of $3$, $55$, $91$ or $95$, there exists a finite extension of degree $d$ for which the Hasse norm principle fails. To accomplish it, we determine the structure of the Tate--Shafarevich groups of norm one tori for finite extensions of degree $d$ under the normality of $p$-Sylow subgroups of the Galois groups of their Galois closures for a square-free prime factor $p$ of $d$. Moreover, we reduce the assertion to an investigation of $2$-dimensional $\mathbb{F}_p$-representations of some groups of order coprime to $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12550 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Hasse norm principle for some non-Galois extensions of square-free degree Oki, Yasuhiro Number Theory In this paper, we study the Hasse norm principle for some non-Galois extensions of number fields. Our main theorem is that for any square-free composite number $d$ which is divisible by at least one of $3$, $55$, $91$ or $95$, there exists a finite extension of degree $d$ for which the Hasse norm principle fails. To accomplish it, we determine the structure of the Tate--Shafarevich groups of norm one tori for finite extensions of degree $d$ under the normality of $p$-Sylow subgroups of the Galois groups of their Galois closures for a square-free prime factor $p$ of $d$. Moreover, we reduce the assertion to an investigation of $2$-dimensional $\mathbb{F}_p$-representations of some groups of order coprime to $p$. |
| title | The Hasse norm principle for some non-Galois extensions of square-free degree |
| topic | Number Theory |
| url | https://arxiv.org/abs/2307.12550 |