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We show that if G$G$ is a finitely generated group such that its profinite completion Ĝ$\widehat{G}$ is "far from being projective" (i.e., the kernel of the universal Frattini cover of Ĝ$\widehat{G}$ is not a small profinite group), then the class of existentially closed G$G$‐actions on fields is not elementary. Since any infinite, finitely generated, virtually free, and not free group is "far from being projective," the main result of this paper corrects an error in our paper, Beyarslan and Kowalski (Proc. London Math. Soc., (2) 118 (2019), 221–256), by showing the negation of Theorem 3.26 in that paper. [ABSTRACT FROM AUTHOR]

There is an irreparable error in the proof of Theorem 3.26 in the above‐mentioned paper and we withdraw the claim of having proved that theorem. In fact, that theorem is false in a very strong sense. [ABSTRACT FROM AUTHOR]

The model companion of the theory of fields with valuation and automorphism (of the pure field structure) exists. A counterexample shows that the theory of models of ACFA equipped with valuation is not this model companion. [ABSTRACT FROM AUTHOR]

We give algebraic conditions for a finite commutative algebra B over a field of positive characteristic, which are equivalent to the companionability of the theory of fields with "B-operators" (i.e., the operators coming from homomorphisms into tensor products with B). We show that, in the most interesting case of a local B, these model companions admit quantifier elimination in the "smallest possible" language, and they are strictly stable. We also describe the forking relation there. [ABSTRACT FROM AUTHOR]

For a group G, we define the notion of a G‐kernel and show that the properties of G‐kernels are closely related with the existence of a model companion of the theory of Galois actions of G. Using Bass–Serre theory, we show that this model companion exists for virtually free groups generalizing the existing results about free groups and finite groups. We show that the new theories we obtain are not simple and not even NTP2. [ABSTRACT FROM AUTHOR]