Prerequisites

Both MATH GU4041 and MATH GU4042, or the equivalent.

This includes: Groups, homomorphisms, normal subgroups, the isomorphism theorems, symmetric groups, group actions, the Sylow theorems, finitely generated abelian groups, rings, homomorphisms, ideals, integral and Euclidean domains, the division algorithm, principal ideal and unique factorization domains, fields, algebraic and transcendental extensions, splitting fields, finite fields, Galois theory.

I advise anyone who did not take Modern Algebra I and II against from taking this course or, at least, to self-teach yourself the materials before the start of the semester.

What this course is about

The primary source will be my lecture notes, which can be found here.

We first aim to develop the basics of algebraic number theory. While doing so, we try to tie into the classical developments of number theory. After that, we focus more on exposing ourselves to modern developments of number theory.

There are various sources for further reference:

Pierre Samuel, Algebraic Theory of Numbers;

Lecture notes of a course taught by Brian Conrad;

James Milne, Algebraic Number Theory;

Hermann Weyl, Algebraic Theory of Numbers (written in old language - still a classic);

Daniel Marcus, Number Fields;

Lecture notes of a course taught by Michael Harris.

References for some of the more advanced topics (which may or may not be dealt later in the course):

Gerald Janusz, Algebraic Number Fields;

Chapters 1~6 of Lawrence Washington, Introduction to Cyclotomic Fields;

Lecture notes of a course taught by Akshay Venkatesh (on geometry of numbers).

Grading

There will be weekly homework assignments (50%), in-class midterm (20%), and a take-home final exam (30%).

Topics

Basic commutative algebra, Number fields, Ring of integers, Finiteness of class number, Dirichlet's unit theorem, Ramification, Local fields, Cyclotomic fields, Statements of local and global class field theory, Dirichlet $L$-functions, Analytic class number formula, Binary quadratic forms.

The main source of the materials for this course is this lecture note.