Towers of totally real nested square roots: undecidability, the lattice of subfields, and the quartic extensions within the tower

Abstract: After recalling some first order undecidability results in infinite algebraic extensions of the field of rational numbers, I will talk about a concrete family of 2-towers of totally real number fields, namely, $(\mathbb{Q}(x_n))_{n\ge0}$, where $x_{n+1}=\sqrt{\nu+x_n}$ for some given positive integers $\nu$ and $x_0$. Let $K$ be the union of the $\mathbb{Q}(x_n)$. Though these fields $K$ are somewhat the simplest subfields of an algebraic closure of $\mathbb{Q}$ that one may construct, they hide a rich variety of natural problems of topological, algebraic, dynamical and logical nature. The results that I will present about these fields are due to M. Castillo, C. Videla, and who writes.

commutative algebralogicnumber theory

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic