Algebraic Number Theory

Instructors:	Daniele Agostini (Lectures), Clemens Nollau (Exercises).
Lectures:	Mo. 12:15-13:45, Mi. 12:15-13:45. Lecture Room N14 (C-Building Morgenstelle).
Exercises:	Tu. 08:15-09:45. Lecture Room N15 (C-Building Morgenstelle). Tu. 16:15-17:45. Seminar Room C2A17 (C-Building Morgenstelle) (only from April 28).
Language:	English.
Registration:	URM.
Lecture notes:	Lectures.

Announcements

Starting from April 28, we are going to have two exercise groups for the course. Please submit your preference on URM by April 21.
Please register on URM for the exercise sessions. The registration is open from April 1st to April 17th, 2026.

Course Description

The course will be a first introduction to Algebraic Number Theory, arguably one of the most important and beautiful subjects in Mathematics. In particular, we will study number fields and their rings of integers.

Prerequisites

Basic abstract algebra, at the level of the courses Linear Algebra I and Linear Algebra II/Algebraic Structures. Knowledge of Galois theory will be useful, but we will repeat the basics at the beginning of the course.

Course material

Notes for the lectures can be found here.

Some handwritten notes of a previous iteration of this course here.

We will not follow exactly any particular book, but the main inspirations for the course will be

D. Marcus, Number fields. Springer.
P. Samuel, Algebraic theory of numbers. Dover.
J. Milne, Algebraic Number Theory. Online notes.

There are many other beautiful references for Algebraic Number Theory. One of them, with a more advanced presentation than the references above, is:

J. Neukirch, Algebraische Zahlentheorie. Springer.

Course Log

A brief summary of each lecture, together with references to the notes, will appear below. The presentation in the notes might be more general and more detailed than the one in the lectures, in particular there are some starred sections in the notes that are meant to be optional, even if we might discuss some of them in the exercise sessions.

13.04.26: Presentation of the course. Primes that are a sum of two squares. (Notes: Chapter 1).
14.04.26: Review of notions from Algebraic Structures: rings, ideals, homomorphisms, Noetherian rings, factorization and divisibility, Euclidean domains, principal ideal domains, unique factorization domains. (Notes: Sections A.1-A.4.4).
15.04.26: Factorization of ideals in a PID. Modules. (Notes: Sections A.4.5-A.5, up to A.5.1)
20.04.26: Finitely generated and free modules. Finitely generated modules over a PID. Cayley-Hamilton theorem. (Notes: Sections A.5.1-A.5.3)

Exam Requirements

There are no formal requirements to access the exam, but students are strongly encouraged to try to solve the exercise sheets and to take active part in the lectures and the exercise sessions.

Exercise Sheets

The exercise sheets will appear below. If you want your exercise sheets to be corrected, you should upload them to URM by the deadline.

Sheet 1. Deadline: 27.04.26.