A Mysterious Ring

Alexandra Shlapentokh (East Carolina University)

28-Oct-2021, 18:00-19:00 (3 years ago)

Abstract: Let Qab{\mathbb Q}^{\text{ab}} be the largest abelian extension of Q\mathbb Q, or in other words the compositum of all cyclotomic extensions. Let OQabO_{{\mathbb Q}^{\text{ab}}} be the ring of integers of Qab{\mathbb Q}^{\text{ab}} or the ring of elements of Qab{\mathbb Q}^{\text{ab}} satisfying monic irreducible polynomials over Z\mathbb Z. It is not known whether the first-order theory of OQabO_{{\mathbb Q}^{\text{ab}}} is decidable. Qab{\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 Qab{\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

Organizer: Wesley Calvert*
*contact for this listing

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