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.

In the latter part of the course, when we learn about zeta functions and L-functions, we will use elementary complex analysis (mostly the notion of holomorphic functions and contour integrals). It will be helpful (but not necessary) to know the basics of complex analysis.

What this course is about

The primary source will be my lecture note/textbook, An Invitation To Modern Algebraic Number Theory.

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.

Other references with slightly different emphasis/perspective that might be helpful for your learning:

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.

Grading

There will be weekly homework assignments (45%), Midterm (20%), Final (20%), and an individual 15-minute presentation (15%).

Both exams are in-person. The Midterm will be held in-class on 3/11, the last class before the Spring Recess.

The Final is projected to be held on 5/11 at 4:10-7pm.

You will be asked to give a 15-minute presentation on anything a step further than an advanced topic taught in this course (e.g., class field theory, arithmetic of cyclotomic fields). Most likely you will choose a part of a paper from a list that I will suggest, but you are also welcome to come up with your own. The date of the presentation is TBD.

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 "textbook" below refers to this document.