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

TL;DR

This paper extends previous results to define algebraic integers in broad classes of infinite algebraic extensions of Q and demonstrates their first-order undecidability, providing structural conditions for definability of the ring of integers.

Contribution

It introduces new definability results for algebraic integers in infinite extensions of Q and establishes undecidability of their first-order theories under certain conditions.

Findings

Algebraic integers are definable in Galois extensions with degrees not divisible by certain primes.

Definability of algebraic integers in cyclotomic extensions generated by specific roots of unity.

First-order theories of certain abelian extensions with finitely many ramified primes are undecidable.

Abstract

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.