Definability of the ring of integers in some infinite algebraic extensions of the rationals (Q2910987)

The paper under review shows first-order definability (in the language of rings) of the rings of algebraic integers over infinite Galois extensions of \(\mathbb Q\) such that every finite subextension is of odd degree over \(\mathbb Q\) and dyadic prime ideals are not ramified. In particular the author shows that the ring of integers is definable over the field \(\mathbb Q(\{\cos 2\pi/\ell^n: \ell \in \triangle, n \in \mathbb N\})\), where \(\triangle\) consists of all the prime numbers congruent to \(-1\) mod 4. Combining this definability result with undecidability results of Julia Robinson concerning undecidability of the first-order theory in totally real rings of algebraic integers, the author proves that the first-order theory of this infinite extension of \(\mathbb Q\) is undecidable. The definability and decidability results are generalizations of some results of Carlos Videla who showed that the integers are definable and the first-order theory is undecidable in infinite extensions of \(\mathbb Q\) where the degrees of all finite subextensions are divisible by a fixed finite set of primes only. The proof technique is based on methods used by Robert Rumely to show first-order undecidability of global fields.