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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.12855 |
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| _version_ | 1866909615629795328 |
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| author | Danh, Le Qui Deo, Trinh Thanh |
| author_facet | Danh, Le Qui Deo, Trinh Thanh |
| contents | Let $D$ be a division ring with center $F$, $f(x_1,x_2,\dots, x_m)$ a non-central multilinear polynomial over $F$, and $w(x_1,x_2,\dots,x_m)$ a non-trivial word. In this paper, we investigate conditions under which there exists an element $a \in D$ such that the subfield $F(a)$ generated by $a$ is a maximal subfield of $D$. Specifically, we prove that there always exists an element $a$ in the set \[ \{f(a_1,\dots,a_m)\mid a_1,\dots, a_m\in D \} \cup \{w(a_1,\dots,a_m)\mid a_1,\dots, a_m\in D \backslash \{0\} \} \] such that $F(a)$ is a maximal subfield of $D$. This result shows that maximal subfields can be generated by evaluating polynomial or group word expressions at elements of $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12855 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal subfields in division algebras generated by images of polynomials Danh, Le Qui Deo, Trinh Thanh Rings and Algebras Let $D$ be a division ring with center $F$, $f(x_1,x_2,\dots, x_m)$ a non-central multilinear polynomial over $F$, and $w(x_1,x_2,\dots,x_m)$ a non-trivial word. In this paper, we investigate conditions under which there exists an element $a \in D$ such that the subfield $F(a)$ generated by $a$ is a maximal subfield of $D$. Specifically, we prove that there always exists an element $a$ in the set \[ \{f(a_1,\dots,a_m)\mid a_1,\dots, a_m\in D \} \cup \{w(a_1,\dots,a_m)\mid a_1,\dots, a_m\in D \backslash \{0\} \} \] such that $F(a)$ is a maximal subfield of $D$. This result shows that maximal subfields can be generated by evaluating polynomial or group word expressions at elements of $D$. |
| title | Maximal subfields in division algebras generated by images of polynomials |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2505.12855 |