First Order Decidability and Definability of Integers in Infinite Algebraic Extensions of Rational Numbers

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show first-order undecidability. In particular, we show that the following propositions hold. (1) For any rational prime q and any positive rational integer m, algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by q. (2) Given a prime q, and an integer m > 0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set {ξpl |l ∈ Z>0, p 6= q is any prime such that q 6 |(p− 1)}. (3) The first-order theory of any abelian extension of Q with finitely many ramified rational primes is undecidable.