Small generators of abelian number fields
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910344821080064 |
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| author | Widmer, Martin |
| author_facet | Widmer, Martin |
| contents | We show that for each abelian number field $K$ of sufficiently large degree $d$ there exists an element $α\in K$ with $K=\IQ(α)$ and absolute Weil height $H(α)\ll_d |Δ_K|^{1/2d}$ , where $Δ_K$ denotes the discriminant of $K$. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent $1/2d$ is best-possible when $d$ is even. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02044 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Small generators of abelian number fields Widmer, Martin Number Theory 11R04, 11G50, 11R29, 11R06 We show that for each abelian number field $K$ of sufficiently large degree $d$ there exists an element $α\in K$ with $K=\IQ(α)$ and absolute Weil height $H(α)\ll_d |Δ_K|^{1/2d}$ , where $Δ_K$ denotes the discriminant of $K$. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent $1/2d$ is best-possible when $d$ is even. |
| title | Small generators of abelian number fields |
| topic | Number Theory 11R04, 11G50, 11R29, 11R06 |
| url | https://arxiv.org/abs/2312.02044 |