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# A primitive element theorem for fields with commuting derivations and automorphisms

• Gleb Pogudin
Article

## 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
• 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 $$a \in E$$ 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.

## Keywords

Primitive element Differential field Difference field Fields with operators

## Mathematics Subject Classification

Primary 12H05 12H05 Secondary 12F99

## Notes

### Acknowledgements

The author is grateful to Lei Fu, Alexey Ovchinnikov, Thomas Scanlon, and the referee for their suggestions and helpful discussions. This work has been partially supported by NSF Grants CCF-1564132, CCF-1563942, DMS-1853482, DMS-1853650, and DMS-1760448, by PSC-CUNY Grants #69827-0047 and #60098-0048.

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