Alexandra Shlapentokh

It is shown that for any computable field K and any r.e. degree a there is an r.e. set A of degree a and a field F ≅ K with underlying set A such that the field operations of F (including subtraction and division) are extendible to (total) recursive functions. Further, it is shown that if a and b are r.e. degrees with b ≤ a, there is a 1-1 recursive function $f: \mathbb{Q} \rightarrow \omega$ such that f(Q) ∈ a, f(Z) ∈ b, and the images of the field operations of Q under f can be extended to recursive functions

Let K be a function field of one variable over a constant field C of finite transcendence degree over C. Let M/K be a finite extension and let W be a set of primes of K such that all but finitely many primes of W do not split in the extension M/K. Then there exists a set W' of K-primes such that Hilbert's Tenth Problem is not decidable over $O_{K,W'} = \{x \in K\mid ord_\mathfrak{p} x \geq 0, \forall\mathfrak{p} \notin W'\}$ , and the set (W' $\backslash$ W) ∪ (W $\backslash$ W') is finite. Let K be a function field of one variable over a constant field C finitely generated over Q. Let M/K be a finite extension and let W be a set of primes of K such that all but finitely many primes of W do not split in the extension M/K and the degree of all the primes in W is bounded by b ∈ N. Then there exists a set W' of K-primes such that Z has a Diophantine definition over O K ,W', and the set (W' $\backslash$ W) ∪ (W $\backslash$ W') is finite

We use a generalization of a construction by Ziegler to show that for any field F and any countable collection of countable subsets Ai⊆FAi⊆F, i∈I⊂Z>0i∈I⊂Z>0 there exist infinitely many fields K of arbitrary greater than one transcendence degree over F and of infinite algebraic degree such that each AiAi is first-order definable over K. We also use the construction to show that many infinitely axiomatizable theories of fields which are not compatible with the theory of algebraically closed fields are finitely hereditarily undecidable

Fried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure${\cal S}$, there exists a countable field${\cal F}$of arbitrary characteristic with the same essential computable-model-theoretic properties as${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our functor and its partially defined inverse are computable functors.

We investigate the issues of Diophantine definability over the non-finitely generated version of non-degenerate modules contained in the infinite algebraic extensions of the rational numbers. In particular, we show the following. Let k be a number field and let K inf be a normal algebraic, possibly infinite, extension of k such that k has a normal extension L linearly disjoint from K inf over k. Assume L is totally real and K inf is totally complex. Let M inf be a non-degenerate O k -module, possibly non-finitely generated and contained in O Kinf . Then M inf contains a submodule M¯ inf such that M inf /M¯ inf is torsion and O k has a Diophantine definition over M¯ inf

Let K be a countable field. Then a weak presentation of K is an isomorphism of K onto a field whose elements are natural numbers, such that all the field operations are extendible to total recursive functions. Given a pair of two non-finitely generated countable fields contained in some overfield, we investigate under what circumstances the overfield has a weak presentation under which the given fields have images of arbitrary Turing degrees or, in other words, we investigate Turing separability of various pairs of non-finitely generated fields

Let F be a finitely generated field and let j : F → N be a weak presentation of F, i.e. an isomorphism from F onto a field whose universe is a subset of N and such that all the field operations are extendible to total recursive functions. Then if R1 and R2 are recursive subrings of F, for all weak presentations j of F, j is Turing reducible to j if and only if there exists a finite collection of non-constant rational functions {Gi} over F such that for every x ε R1 for some i, Gi ε R2. We investigate under what circumstances such a collection of rational functions exists and conclude that in the case when R1 R2 are both holomorphy rings and F is of characteristic 0 or is an algebraic function field over a perfect field of constants, the existence of the above-described collection of rational functions is equivalent to the requirement that the non-archimedean primes which do not appear as poles of elements of R2 do not have factors of relative degree 1 in some simple extension of K