# Primitive Element Theorem for Fields with Commuting Derivations and Automorphisms

## TL;DR

This paper generalizes the Primitive Element Theorem to fields with multiple commuting derivations and automorphisms, showing such fields can be generated by a single element under broad conditions.

## Contribution

It extends previous theorems by allowing any number of derivations and automorphisms without restrictions on the base field, broadening the theorem's applicability.

## Key findings

- Established a Primitive Element Theorem for fields with multiple commuting operators.
- Proved that such fields can be generated by a single element under specified conditions.
- Generalized prior results by Kolchin and Cohn to more complex operator settings.

## Abstract

We establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism. More precisely, we show that for every extension $F \subset E$ of such fields of zero characteristic such that $\bullet$ $E$ is generated over $F$ by finitely many elements using the field operations and the operators, $\bullet$ every element of $E$ satisfies a nontrivial equation with coefficient in $F$ involving the field operations and the operators, $\bullet$ the action of the operators on $E$ is irredundant there exists an element $a \in E$ such that $E$ is generated over $F$ by $a$ using the field operations and the operators. This result generalizes the Primitive Element Theorems by Kolchin and Cohn in two directions simultaneously: we allow any numbers of derivations and automorphisms and do not impose any restrictions on the base field $F$.