Rings of Integers & 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 \(\mathbb{N}_0[\sqrt{2}]\).
The course consisted of weekly meetings — two lectures and one discussion session. Lectures presented the main content; discussion sessions focused on concrete examples. The course ran from Tuesday, April 28, 2026 to Thursday, June 18, 2026.
Tuesday & Thursday, 7:30–8:30pm ET
Sunday, 5:00–6:00pm ET
- 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