B3.4 Algebraic Number Theory (2022-23)

General Prerequisites:
Rings and Modules and Number Theory. B3.1 Galois Theory is an essential pre-requisite. All second-year algebra and arithmetic. Students who have not taken Part A Number Theory should read about quadratic residues in, for example, the appendix to Stewart and Tall. This will help with the examples.
Course Term: Hilary
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: H
Assessment Type: Written Examination
Course Overview:
An introduction to algebraic number theory. The aim is to describe the properties of number fields, but particular emphasis in examples will be placed on quadratic fields, where it is easy to calculate explicitly the properties of some of the objects being considered. In such fields the familiar unique factorisation enjoyed by the integers may fail, and a key objective of the course is to introduce the class group which measures the failure of this property.
Learning Outcomes:
Students will learn about the arithmetic of algebraic number fields. They will learn to prove theorems about integral bases, and about unique factorisation into ideals. They will learn to calculate class numbers, and to use the theory to solve simple Diophantine equations.
Course Synopsis:
Field extensions, minimum polynomial, algebraic numbers, conjugates, discriminants, Gaussian integers, algebraic integers, integral basis
Examples: quadratic fields
Norm of an algebraic number
Existence of factorisation
Factorisation in \(\mathbb{Q}(\sqrt{d})\)
Ideals, \(\mathbb{Z}\)-basis, maximal ideals, prime ideals
Unique factorisation theorem of ideals
Relationship between factorisation of number and of ideals
Norm of an ideal
Ideal classes
Statement of Minkowski convex body theorem
Finiteness of class number
Computations of class number to go on example sheets