# MAGIC072: Number Theory

## Course details

A core MAGIC course

### Semester

Autumn 2011
Monday, October 10th to Friday, December 16th

### Hours

Live lecture hours
10
Recorded lecture hours
0
0

Fridays
12:05 - 12:55

## Description

We all know how indispensable it is to extend R to C. Often it is better to start with Q and make a finite extension, to get a number field, but there are many more ways to do this-the Galois group of the algebraic closure of Q is immensely complicated. The elements of a number field are algebraic numbers, satisfying monic polynomial equations with coefficients in Q. Among these are the algebraic integers, satisfying monic polynomial equations with coefficients in Z, and they form a subring, the ring of integers. When the number field is Q, this subring is just Z.
This course is about number fields and especially their rings of integers. In general these are not unique factorisation domains, but we shall see how unique factorisation can be restored by using ideals rather than elements. We are led naturally to consideration of the ideal class group and the unit group. This is not just a branch of algebra. We shall use also the geometry of numbers, and some analytic functions, recognising the fact that an algebraic number can be thought of as an element of C, and has a size.

### Prerequisites

Elementary number theory (primes, linear congruences). Rings and groups, including irreducibles, units, Euclidean domains, quotient groups, and preferably the quotient of a ring by an ideal, and finitely generated abelian groups. Field extensions, including the degree of an extension, irreducible polynomials, and preferably Eisenstein's criterion. Galois theory may be mentioned in passing, or in an exercise, but is not essential.

### Syllabus

Finite extensions of Q. Norms, traces and discriminants.
Proof that the subset of algebraic integers is a subring. Existence of an integral basis for the ring of integers.
Ideals, principal ideals, the ideal class group and its finiteness.
Unique factorisation of ideals.
Explicit factorisation of rational primes in rings of integers of number fields.
Minkowski's constant, calculating the ideal class group. Quadratic examples and applications to diophantine equations.
Units. The logarithmic embedding, statement of Dirichlet's unit theorem.
Formula for counting ideals of bounded norm, rough idea of proof. Dedekind zeta function and Dirichlet's class number formula.

## Lecturer

• ND

### Dr Neil Dummigan

University
University of Sheffield

## Bibliography

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

## Assessment

The assessment for this course will be released on Monday 21st September 2020 and is due in by Tuesday 10th January 2012 at 23:59.

There will be a single assignment, with one big question, which has to be passed in order to pass the module. It will be set on 9th December 2011, and due in at noon on 10th January 2012.

Please note that you are not registered for assessment on this course.