Introductory description

Algebraic number theory is the study of algebraic numbers and especially algebraic integers, which are the roots of monic polynomials with integer coefficients. We study ways in which properties of algebraic integers resemble or differ from the properties of ordinary integers, especially with regard to prime factorisation. As well as their intrinsic elegance, this theory of algebraic integers leads to methods for finding solutions of equations in ordinary integers, such as the Mordell equation and Pell equation. Historically, much of the theory was developed in attempts to solve Fermat's Last Theorem. The module combines substantial theoretical results, relying on algebraic tools such as rings, ideals and abelian groups, with calculations of specific examples.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The objects of study in MA3A6 are number fields (finite field extensions of the rational numbers) and rings of algebraic integers. After defining these objects and their basic properties, we focus on factorisation in rings of integers. Unique factorisation into irreducible elements often fails in these rings, but there is unique factorisation for ideals into prime ideals - this is one of the biggest theorems in the module. We use ideals to define the class group of a number field, a finite abelian group which measures how badly unique factorisation fails. We study methods for calculating the class group and the prime factorisation of ideals, and apply these tools to solve some Diophantine equations.

Learning outcomes

By the end of the module, students should be able to:

• be able to compute norms and discriminants and to use them to determine the integer rings in algabraic number fields;
• be able to factorize ideals into prime ideals in algebraic number fields in straightforward examples;
• understand the proof of Minkowski's Theorem on lattices, and be able to apply it, for example, to prove that all positive integers are the sum of four squares.

Indicative reading list

This module is based on the book Algebraic Number Theory and Fermat's Last Theorem, by I.N. Stewart and D.O. Tall, published by A.K. Peters (2001). The contents of the module forms a proper subset of the material in that book. (The earlier edition, published under the title Algebraic Number Theory, is also suitable.) Another recommended book is Algebraic Number Theory by F. Jarvis (Springer). More advanced books are A Brief Guide to Algebraic Number Theory, by H.P.F. Swinnerton-Dyer (LMS Student Texts # 50, CUP), or Algebraic Number Theory, by A. Fröhlich and M.J. Taylor (CUP).

Subject specific skills

In this module, students will acquire a foundation for further study in number theory and for understanding its applications in cryptography. They will also develop skills to link abstract mathematical theory with computation in special cases.

Transferable skills

In this module, students improve their problem solving and analytical skills, with an emphasis on applying theoretical knowledge in computational settings. Students also develop their skills in clearly communicating logical arguments.