Small integral generators of totally complex number fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915444908097536 |
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| author | Akhtari, Shabnam Vaaler, Jeffrey Widmer, Martin |
| author_facet | Akhtari, Shabnam Vaaler, Jeffrey Widmer, Martin |
| contents | Let $K$ be an algebraic number field and $H$ the absolute Weil height. Write $c_K$ for a certain positive constant that is an invariant of $K$. We consider the question: does $K$ contain an algebraic integer $α$ such that both $K = \mathbb{Q}(α)$ and $H(α) \le c_K$? If $K$ has a real embedding then a positive answer was established in previous work. Here we obtain a positive answer if $\textrm{Tor}\bigl(K^{\times}\bigr) \not= \{\pm 1\}$, and so $K$ has only complex embeddings. We also show that if the answer is negative, then $K$ is totally complex, $\textrm{Tor}\bigl(K^{\times}\bigr) = \{\pm 1\}$, and $K$ is a Galois extension of its maximal totally real subfield. Further, we show that if $μ\in O_K$ is not totally real, then there exists $α$ in $O_K$ with $K = \mathbb{Q}(α)$ and $H(α) \le H(μ)\thinspace c_K$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_11849 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Small integral generators of totally complex number fields Akhtari, Shabnam Vaaler, Jeffrey Widmer, Martin Number Theory 11H06, 11R29, 11R56 Let $K$ be an algebraic number field and $H$ the absolute Weil height. Write $c_K$ for a certain positive constant that is an invariant of $K$. We consider the question: does $K$ contain an algebraic integer $α$ such that both $K = \mathbb{Q}(α)$ and $H(α) \le c_K$? If $K$ has a real embedding then a positive answer was established in previous work. Here we obtain a positive answer if $\textrm{Tor}\bigl(K^{\times}\bigr) \not= \{\pm 1\}$, and so $K$ has only complex embeddings. We also show that if the answer is negative, then $K$ is totally complex, $\textrm{Tor}\bigl(K^{\times}\bigr) = \{\pm 1\}$, and $K$ is a Galois extension of its maximal totally real subfield. Further, we show that if $μ\in O_K$ is not totally real, then there exists $α$ in $O_K$ with $K = \mathbb{Q}(α)$ and $H(α) \le H(μ)\thinspace c_K$. |
| title | Small integral generators of totally complex number fields |
| topic | Number Theory 11H06, 11R29, 11R56 |
| url | https://arxiv.org/abs/2307.11849 |