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

Caleb Springer

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arXiv · 1910.01239

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

Caleb Springer

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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 $d \geq 2$.

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Caleb Springer. 2019-10-02. Undecidability, unit groups, and some totally imaginary infinite extensions of $\mathbb{Q}$. https://arxiv.org/abs/1910.01239

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