Undecidability, unit groups, and some totally imaginary infinite extensions of ℚ

Springer, Caleb

doi:10.1090/proc/15153

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

Proc. Amer. Math. Soc. 148 (2020), 4705-4715

DOI: https://doi.org/10.1090/proc/15153

Published electronically: August 11, 2020

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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 Martínez-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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