A 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 ⊂ E of such fields of zero characteristic such that • E 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 .

Model Theory of Derivations of the Frobenius Map Revisited

“…It seems that idea of the proof of Theorem 3.13 can be applied in a more general context. Whenever we have a theory of fields with some operators satisfying some primitive element theorem (e.g., as in [8]), then -existential closedness is equivalent to existential closedness. On the other hand, if a theory of fields has a model companion which eliminates quantifiers (in some reasonable language L ), then it is easy to see that the class of -existentially closed models of in the language L is elementary.…”

Model theory of derivations of the Frobenius map revisited

“…It seems that idea of the proof of Theorem 3.12 can be applied in a more general context. Whenever we have a theory T ′ of fields with some operators satisfying some primitive element theorem (e. g. as in [6]), then 1-existential closedness is equivalent to existential closeness. On the other hand, if a theory of fields T ′ has a model companion which eliminates quantifiers (in some reasonable language L), then it is easy to see that the class of 1-existentially closed models of T ′ in the language L is elementary.…”