Course Catalogue

View Specification Document | Reading List

This course introduces the foundations of algebraic number theory as developed by Kummer and Dedekind in the early 20th century. It teaches students algebraic approaches to solving Diophantine equations.

2 lectures per week for 11 weeks, 1 tutorial per week for 10 weeks

Mandatory:

■ basic ring theory, as taught, e.g., in 3H: Algebra, MATHS4072

Recommended:

■ elementary number theory, as taught, e.g., in 4H Number Theory, MATHS4108

Galois theory

N/A

N/A

180min in April-May

The aim of the course is to introduce the students to algebraic number fields, their rings of integers, to the problem of unique factorisation and to ideal class groups, and to applications of this theory to Diophantine equations.

By the end of this course students will be able to:

■ define the notions of an algebraic number field, of its ring of integers, and of its class group;

■ understand the hierarchy between UFDs, PIDs, Euclidean Domains, and define the notion of a Dedekind domain, as well as give examples of rings at various points in this hierarchy;

■ compute norms and traces of algebraic numbers, the discriminant of an order, the ring of integers of a number field and its unit group in concrete examples;

■ compute factorisations of ideals in rings of integers;

■ apply geometry of numbers to compute the class group of a ring of integers in concrete examples;

■ state and prove the main theorems concerning the above invariants;

find integral solutions to certain cubic polynomial equations using class groups.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.