Strongly Normal Extensions and Algebraic Differential Equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913953019330560 |
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| author | Kumbhakar, Partha Srinivasan, Varadharaj Ravi |
| author_facet | Kumbhakar, Partha Srinivasan, Varadharaj Ravi |
| contents | Let $k$ be a differential field having an algebraically closed field of constants, $E$ be a strongly normal extension of $k$, and $k^0$ be the algebraic closure of $k$ in $E.$ We prove for any intermediate differential field $k\subset K\subseteq E$ that there is an intermediate differential field $k\subset M\subseteq K$ such that either $M$ is generated as a differential field over $k$ by a nonalgebraic solution of a Riccati differential equation over $k$ or $k^0M$ is an abelian extension of $k^0$. Using this result, we reprove and extend certain results of Goldman and Singer and study $d-$solvability of linear differential equations. We also extend a result of Rosenlicht and study algebraic dependency of solutions of algebraic differential equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16435 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly Normal Extensions and Algebraic Differential Equations Kumbhakar, Partha Srinivasan, Varadharaj Ravi Commutative Algebra 12H05, 12H20, 14LXX Let $k$ be a differential field having an algebraically closed field of constants, $E$ be a strongly normal extension of $k$, and $k^0$ be the algebraic closure of $k$ in $E.$ We prove for any intermediate differential field $k\subset K\subseteq E$ that there is an intermediate differential field $k\subset M\subseteq K$ such that either $M$ is generated as a differential field over $k$ by a nonalgebraic solution of a Riccati differential equation over $k$ or $k^0M$ is an abelian extension of $k^0$. Using this result, we reprove and extend certain results of Goldman and Singer and study $d-$solvability of linear differential equations. We also extend a result of Rosenlicht and study algebraic dependency of solutions of algebraic differential equations. |
| title | Strongly Normal Extensions and Algebraic Differential Equations |
| topic | Commutative Algebra 12H05, 12H20, 14LXX |
| url | https://arxiv.org/abs/2507.16435 |