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Register a new userOn algebraic closure in pseudofinite fields
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We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F. We show that the behavior of this group, even when A is large, depends essentially on the roots of unity in F. For almost all completions of the theory of pseudo-finite fields we show that algebraic closure agrees with definable closure, as soon as A contains the relative algebraic closure of the prime field.
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This second part of the paper strengthens the descent theory described in the first part to rational maps, arbitrary base fields, and dynamics given by correspondences. We obtain in particular a decomposition of any difference field extension into a tower of finite, field-internal and one-based difference field extensions. This is needed in order to obtain the dynamical Northcott Theorem 1.11 of Part I in sharp form.
In several classes of countable structures it is known that every hyperarithmetic structure has a computable presentation up to bi-embeddability. In this article we investigate the complexity of embeddings between bi-embeddable structures in two such classes, the classes of linear orders and Boolean algebras. We show that if $mathcal L$ is a computable linear order of Hausdorff rank $n$, then for every bi-embeddable copy of it there is an embedding computable in $2n-1$ jumps from the atomic diagrams. We furthermore show that this is the best one can do: Let $mathcal L$ be a computable linear order of Hausdorff rank $ngeq 1$, then $mathbf 0^{(2n-2)}$ does not compute embeddings between it and all its computable bi-embeddable copies. We obtain that for Boolean algebras which are not superatomic, there is no hyperarithmetic degree computing embeddings between all its computable bi-embeddable copies. On the other hand, if a computable Boolean algebra is superatomic, then there is a least computable ordinal $alpha$ such that $mathbf 0^{(alpha)}$ computes embeddings between all its computable bi-embeddable copies. The main technique used in this proof is a new variation of Ash and Knights pairs of structures theorem.
After recalling the definition of Zilber fields, and the main conjecture behind them, we prove that Zilber fields of cardinality up to the continuum have involutions, i.e., automorphisms of order two analogous to complex conjugation on (C,exp). Moreover, we also prove that for continuum cardinality there is an involution whose fixed field, as a real closed field, is isomorphic to the field of real numbers, and such that the kernel is exactly 2{pi}iZ, answering a question of Zilber, Kirby, Macintyre and Onshuus. The proof is obtained with an explicit construction of a Zilber field with the required properties. As further applications of this technique, we also classify the exponential subfields of Zilber fields, and we produce some exponential fields with involutions such that the exponential function is order-preserving, or even continuous, and all of the axioms of Zilber fields are satisfied except for the strong exponential-algebraic closure, which gets replaced by some weaker axioms.
We prove that for every Scott set $S$ there are $S$-saturated real closed fields and models of Presburger arithmetic.
Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. Let $K$ be a regular field which is not generically stable and let $p$ be its global generic type. We observe that if $K$ has a finite extension $L$ of degree $n$, then $p^{(n)}$ has unbounded orbit under the action of the multiplicative group of $L$. Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique non-trivial conjugacy class, and we notice that a classical group with one non-trivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then we construct a group of cardinality $omega_1$ with only one non-trivial conjugacy class and such that the centralizers of all non-trivial elements are countable.
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