标题： 强正规扩张与代数微分方程 显示英文标题

标题： Strongly Normal Extensions and Algebraic Differential Equations

摘要： 设$k$是一个常数域为代数闭域的微分域，$E$是$k$的强正规扩张，而$k^0$是$k$在$E.$中的代数闭包。我们证明对于任何中间微分域$k\subset K\subseteq E$，存在一个中间微分域$k\subset M\subseteq K$，使得$M$或者是由 Riccati 微分方程在$k$上的一个非代数解在$k$上生成的微分域，或者$k^0M$是$k^0$的阿贝尔扩张。 利用这个结果，我们重新证明并扩展了Goldman和Singer的某些结果，并研究了线性微分方程的$d-$可解性。我们还扩展了Rosenlicht的一个结果，并研究了代数微分方程解的代数相关性。 显示英文摘要

摘要： 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.