Generic multiplicative endomorphism of a field

TL;DR

This paper introduces a new model companion for fields expanded by a multiplicative endomorphism, analyzing its model-theoretic properties such as NSOP1, simplicity, and elimination of imaginaries.

Contribution

It establishes the theory ACFH for fields with a multiplicative endomorphism and explores its model-theoretic properties, including NSOP1 and conditions for elimination of imaginaries.

Findings

ACFH is NSOP1 and not simple

Kernel of the multiplicative map is a pseudo-finite abelian group

Elimination of imaginaries holds if forking satisfies existence

Abstract

We introduce the model-companion of the theory of fields expanded by a unary function for a multiplicative map, which we call ACFH. Among others, we prove that this theory is NSOP$_1$ and not simple, that the kernel of the map is a generic pseudo-finite abelian group. We also prove that if forking satisfies existence, then ACFH has elimination of imaginaries.