Universal quadratic forms and Northcott property of infinite number fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929602690023424 |
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| author | Daans, Nicolas Kala, Vítězslav Man, Siu Hang |
| author_facet | Daans, Nicolas Kala, Vítězslav Man, Siu Hang |
| contents | We show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals $\mathbb{Q}$, then the set of totally positive integers in the extension does not have the Northcott property. In particular, this implies that no universal form exists over the compositum of all totally real Galois fields of a fixed prime degree over $\mathbb{Q}$. Further, by considering the existence of infinitely many square classes of totally positive units, we show that no classical universal form exists over the compositum of all such fields of degree $3d$ (for each fixed odd integer $d$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_16721 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Universal quadratic forms and Northcott property of infinite number fields Daans, Nicolas Kala, Vítězslav Man, Siu Hang Number Theory 11E12 (Primary) 11E20, 11G50, 11H55, 11R04, 11R20, 11R80 (Secondary) We show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals $\mathbb{Q}$, then the set of totally positive integers in the extension does not have the Northcott property. In particular, this implies that no universal form exists over the compositum of all totally real Galois fields of a fixed prime degree over $\mathbb{Q}$. Further, by considering the existence of infinitely many square classes of totally positive units, we show that no classical universal form exists over the compositum of all such fields of degree $3d$ (for each fixed odd integer $d$). |
| title | Universal quadratic forms and Northcott property of infinite number fields |
| topic | Number Theory 11E12 (Primary) 11E20, 11G50, 11H55, 11R04, 11R20, 11R80 (Secondary) |
| url | https://arxiv.org/abs/2308.16721 |