Undecidability, unit groups, and some totally imaginary infinite extensions of $\mathbb{Q}$

TL;DR

This paper constructs new examples of totally imaginary infinite extensions of the rationals with undecidable first-order theories, extending previous methods to a broader class of fields.

Contribution

It generalizes existing techniques to prove undecidability for a wide range of totally imaginary infinite extensions of , including ^{(d)}_{ab} for all d  2.

Findings

Proves undecidability of ^{(d)}_{ab} for all d  2.

Introduces parametrized polynomials with totally real units.

Extends methods from totally real to totally imaginary fields.

Abstract

We produce new examples of totally imaginary infinite extensions of $\mathbb{Q}$ which have undecidable first-order theory by generalizing the methods used by Martinez-Ranero, Utreras and Videla for $\mathbb{Q}^{(2)}$. In particular, we use parametrized families of polynomials whose roots are totally real units to apply methods originally developed to prove the undecidability of totally real fields. This proves the undecidability of $\mathbb{Q}^{(d)}_{ab}$ for all $d \geq 2$.