Introduction to Algebraic Number Theory

Introduction to Algebraic Number Theory 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

The course is led by Dr. Felix Gotti and assisted by Dr. Harold Polo and Victor Gonzalez.

The class meets every Tuesday and Thursday, starting on Tuesday, April 28, 2026 and ending on Thursday, June 18, 2026, via Zoom.

To join the class, please use the following Zoom link: https://mit.zoom.us/j/94236513729.

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

Schedule

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

Back to top