Strongly Normal Extensions and Algebraic Differential Equations

TL;DR

This paper investigates the structure of strongly normal differential field extensions, showing how intermediate fields relate to solutions of Riccati equations or abelian extensions, and applies these findings to differential equations and algebraic dependencies.

Contribution

It establishes a new structural theorem for intermediate fields in strongly normal extensions, extending previous results and analyzing algebraic dependencies of solutions.

Findings

Existence of intermediate fields generated by Riccati solutions or abelian extensions.

Reproof and extension of Goldman and Singer's results.

Insights into d-solvability and algebraic dependencies of differential equations.

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.