# On the undecidability of infinite algebraic extensions of the rationals

## Algebra and Number Theory Seminar

## Meeting Details

For more information about this meeting, contact Allyson Borger, Kirsten Eisentraeger, Jack Huizenga, Mihran Papikian, Ae Ja Yee, John Lesieutre.

**Speaker:** Caleb Springer, PSU

**Abstract:** A ring $R$ is said to be decidable if there is an algorithm which decides whether arbitrary first-order formulas hold in $R$, and $R$ is undecidable otherwise. Julia Robinson proved that $\mathbb{Q}$ is undecidable, and extended this result to all finite extensions of $\mathbb{Q}$. However, the situation gets more complicated for infinite algebraic extensions of $\mathbb{Q}$. Examples show that some infinite algebraic extensions are undecidable, while others are not, and results remain unknown in general. This talk will give an overview of the known results, and present new examples of undecidable totally imaginary infinite extensions of $\mathbb{Q}$ by leveraging unit groups and methods developed for totally real fields.

## Room Reservation Information

**Room Number:** 106 McAllister

**Date:** 10/03/2019

**Time:** 11:00am - 11:50am