On closed rational functions in several variables
Let \(\mathbb K= \bar{\mathbb K}\) be a field of characteristic zero. An element \(\varphi\in \mathbb K(x_1,\dots, x_{n})\) is called a closed rational function if the subfield \(\mathbb K(\varphi)\) is algebraically closed in the field \(\mathbb K(x_1,\dots, x_{n})\). We prove that a rational funct...
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| Date: | 2018 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2018
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/848 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | Let \(\mathbb K= \bar{\mathbb K}\) be a field of characteristic zero. An element \(\varphi\in \mathbb K(x_1,\dots, x_{n})\) is called a closed rational function if the subfield \(\mathbb K(\varphi)\) is algebraically closed in the field \(\mathbb K(x_1,\dots, x_{n})\). We prove that a rational function \(\varphi=f/g\) is closed if \(f\) and \(g\) are algebraically independent and at least one of them is irreducible. We also show that a rational function \(\varphi=f/g\) is closed if and only if the pencil \(\alpha f+\beta g\) contains only finitely many reducible hypersurfaces. Some sufficient conditions for a polynomial to be irreducible are given. |
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