Rings of Integers and Beyond

Rings of Integers and Beyond explores the arithmetic of rings of integers, ideals, and related factorization phenomena. It is intended as a rigorous bridge between elementary number theory and abstract algebra, in the same spirit of inquiry that guides the Summer Workshop for Intrepid Mathematicians (SWIM).

It is designed for students who are ready to move beyond standard coursework and begin engaging with research-facing questions in number theory, commutative algebra, semiring theory, and factorization theory.

The material moves from the concrete arithmetic of the Gaussian and Eisenstein rings of integers to the structural theory of Dedekind domains and the geometry of numbers. Although the primary focus is the study of rings of integers \(\mathcal{O}_K\), we will also make occasional excursions into orders, such as \(\mathbb{Z}[\sqrt{5}]\), and monogenic semidomains, such as \(\mathbb{N}_0[\sqrt{2}]\).

Instructors

Lectures and Discussions

The course consists of weekly meetings: two lectures and one discussion session. The main content of the course will be presented during the lectures, while the discussion sessions will focus on concrete examples. The weekly schedule is as follows.

  • Weekly lectures: Tuesday and Thursday, from 7:30pm to 8:30pm ET, via this Zoom link.
  • Discussion sessions: Sunday, from 5:00pm to 6:00pm ET, via this Zoom link.

The course will run from Tuesday, April 28, 2026, to Thursday, June 18, 2026.

Some of the topics to be highlighted include:

  • Fermat’s Last Theorem and the failure of unique factorization in \(\mathbb{Z}[\zeta_{23}]\)
  • Unique factorization in the monoid of ideals of a ring of integers
  • The finite factorization property in \(\mathcal{O}_K\)
  • The half-factorial property and the divisor class group, including Carlitz’s theorem
  • The Davenport constant as the combinatorial engine behind Carlitz’s theorem

Resources

Background Reading. The following preliminary documents provide background material that may be helpful for the course.

Course Notes. These are the main course notes. They will be updated regularly throughout the course.

Schedule

Please see the schedule below for announced lecture dates and titles. The remaining lectures will be announced shortly.

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