Math 784 Algebraic Number Theory
Graduate Course, University of South Carolina, Department of Mathematics, 2024
Course Description: This course aims to provide an introductory exploration into algebraic numbers and algebraic integers. Algebraic number theory extends its focus beyond integers, particularly emphasizing number fields as finite algebraic extensions of Q. The following topics will be covered in the course as time permits: Algebraic Numbers/Integers, Number Fields, Quadratic Fields, Cyclotomic Fields, Dedekind Domains, Orders, Factorization of Ideals, Minkowski’s Theorem, Geometry of Numbers, Ideal Classes, Dirichlet’s Unit Theorem, Splitting of Prime Ideals, Artin Reciprocity, L-functions, Class Number Formulas, and Class Field Theory.