A primitive element theorem for fields with commuting derivations and automorphisms

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⊂ E of such fields of zero characteristic such thatE is generated over F by finitely many elements using the field operations and the operators,every element of E satisfies a nontrivial equation with coefficient in F involving the field operations and the operators,the action of the operators on E is irredundant there exists an element a∈ 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.