Strongly Normal Extensions and Algebraic Differential Equations

Partha Kumbhakar; Varadharaj Ravi Srinivasan

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arXiv · 2507.16435

Strongly Normal Extensions and Algebraic Differential Equations

Partha Kumbhakar·Varadharaj Ravi Srinivasan

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Abstract

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.

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BibTeX RIS

Partha Kumbhakar, Varadharaj Ravi Srinivasan. 2025-07-22. Strongly Normal Extensions and Algebraic Differential Equations. https://arxiv.org/abs/2507.16435

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