A Mysterious Ring

Abstract: Let ${\mathbb Q}^{\text{ab}}$ be the largest abelian extension of $\mathbb Q$, or in other words the compositum of all cyclotomic extensions. Let $O_{{\mathbb Q}^{\text{ab}}}$ be the ring of integers of ${\mathbb Q}^{\text{ab}}$ or the ring of elements of ${\mathbb Q}^{\text{ab}}$ satisfying monic irreducible polynomials over $\mathbb Z$. It is not known whether the first-order theory of $O_{{\mathbb Q}^{\text{ab}}}$ is decidable. ${\mathbb Q}^{\text{ab}}$ is also a degree two extension of a totally real field. Much more is known about the first-order theory of rings of integers of totally real fields and in some cases one is able to deduce undecidability of the first-order theory of the ring of integers of a degree 2 extension of a totally real field from an analogous result for the ring of integers of the totally real field. However this method does not seem to work for ${\mathbb Q}^{\text{ab}}$. We discuss a possible way of resolving this problem and some related questions.

logicnumber theory

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic