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

Alexandra Shlapentokh

Published 2013-07-02, updated 2014-10-22Version 3

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 the first-order undecidability. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. 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 $q^{m}$. (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 $\{\xi_{p^{\ell}}| \ell \in \Z_{>0}, p \not=q {is any prime such that} q^{m +1}\not | (p-1)\}$. (3) The first-order theory of any abelian extension of Q with finitely many ramified rational primes is undecidable. We also show that under a condition on the splitting of one rational prime in an infinite algebraic extension of Q, the existence of a finitely generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is indecidable.