Primitive Element Theorem for Fields with Commuting Derivations and Automorphisms

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 $F$.