Courses / Module

Module Objective: to give a rigorous course in algebraic number theory

Algebraic number theory uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. The course begins with a review of commutative algebra, including modules, Noetherian rings, finitely generated abelian groups and the Chinese remainder theorem. Algebraic numbers and algebraic integers, and the ring of integers in a number field. The ring of integer in a number field is noetherian, integrally closed and of dimension 1. Relative trace and norm maps on a field extension. Ring of integers is an order. Dedekind Domains and unique factorization of ideals in Dedekind Domains. The Class group of a number field is finitely generated, Minkowski's constant and Minkowski's Theorem. Other topics include cyclotomic number fields and Dirichlet's Unit Theorem.