Topics of Study: This course will be an introduction to the study of algebraic numbers and algebraic integers. In number theory the key motivating problem is to understand the basic arithmetic of the integers. Algebraic number theory is the study of generalizations of integers to other domains, especially to number fields, ie, finite algebraic extensions of Q. Interesting problems arise in the study of rings of algebraic integers that shed light on many basic number theory problems. The course will cover the following topics as time permits:

• algebraic numbers and algebraic integers
• rings of algebraic integers
• unique factorization of ideals
• Minkowski's Theorem and the geometry of numbers
• ideal classes and the finiteness of the class number
• Dirichlet's unit theorem
• decomposition group, inertia groups and ramification groups of prime ideals
• quadratic number fields
• cyclotomic number fields
• Dedekind zeta function and class number formulas
• localizations and p-adic numbers

Prerequisites: Students should be familiar with the topics covered in a first-year graduate course in Algebra, including standard results on groups, rings, fields, vector spaces, modules, and Galois theory. No previous background in number theory is necessary.