Linear relations of four conjugates of an algebraic number
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918097156308992 |
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| author | Baronėnas, Ž. Drungilas, P. Jankauskas, J. |
| author_facet | Baronėnas, Ž. Drungilas, P. Jankauskas, J. |
| contents | We characterize all algebraic numbers $α$ of degree $d\in\{4,5,6,7\}$ for which there exist four distinct algebraic conjugates $α_1$, $α_2$, $α_3$, $α_4$ of $α$ satisfying the relation $α_{1}+α_{2}=α_{3}+α_{4}$. In particular, we prove that an algebraic number $α$ of degree 6 satisfies this relation with $α_{1}+α_{2}\notin\mathbb{Q}$ if and only if $α$ is the sum of a quadratic and a cubic algebraic number. Moreover, we describe all possible Galois groups of the normal closure of $\mathbb{Q}(α)$ for such algebraic numbers $α$. We also consider similar relations $α_{1}+α_{2}+α_{3}+α_{4}=0$ and $α_{1}+α_{2}+α_{3}=α_{4}$ for algebraic numbers of degree up to 7. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13831 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear relations of four conjugates of an algebraic number Baronėnas, Ž. Drungilas, P. Jankauskas, J. Number Theory 11R04, 11R32 We characterize all algebraic numbers $α$ of degree $d\in\{4,5,6,7\}$ for which there exist four distinct algebraic conjugates $α_1$, $α_2$, $α_3$, $α_4$ of $α$ satisfying the relation $α_{1}+α_{2}=α_{3}+α_{4}$. In particular, we prove that an algebraic number $α$ of degree 6 satisfies this relation with $α_{1}+α_{2}\notin\mathbb{Q}$ if and only if $α$ is the sum of a quadratic and a cubic algebraic number. Moreover, we describe all possible Galois groups of the normal closure of $\mathbb{Q}(α)$ for such algebraic numbers $α$. We also consider similar relations $α_{1}+α_{2}+α_{3}+α_{4}=0$ and $α_{1}+α_{2}+α_{3}=α_{4}$ for algebraic numbers of degree up to 7. |
| title | Linear relations of four conjugates of an algebraic number |
| topic | Number Theory 11R04, 11R32 |
| url | https://arxiv.org/abs/2507.13831 |