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

TL;DR

This paper extends Kolchin's primitive element theorem for differential fields, showing that if a finitely generated differential extension contains a nonconstant element, it can be generated by a single element.

Contribution

The paper generalizes Kolchin's theorem by removing the requirement that the nonconstant element be in the base field, allowing it to be in the extension.

Findings

The theorem applies to differential fields with zero derivation on the ground field.

Any finitely generated differential extension with a nonconstant element can be generated by one element.

Strengthens the understanding of the structure of differential field extensions.

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 $E$.