The primitive element theorem for differential fields with zero derivation on the ground field

Abstract

In this paper we strengthen Kolchin's theorem ([1]) in the ordinary case. It states that if a differential field $E$ is finitely generated over a differential subfield $F \subset E$, $trdeg_F E < \infty$, and $F$ contains a nonconstant, i.e. an element $f$ such that $f^{\prime} \neq 0$, then there exists $a \in E$ such that $E$ is generated by $a$ and $F$. We replace the last condition with the existence of a nonconstant element in $E$.