Strongly Normal Extensions and Algebraic Differential Equations

Let k be a differential field having an algebraically closed field of constants, E be a strongly normal extension of k, and k⁰ be the algebraic closure of k in $E.$ We prove for any intermediate differential field k⊂ K⊆ E that there is an intermediate differential field k⊂ M⊆ 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 k⁰M is an abelian extension of k⁰. 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.