Algebraic Number Theory (LTCC, January-February 2009)

Algebraic Number Theory, January-February 2009

For PhD students at the London Taught Course Centre.

This course gives a non-technical introduction to some important topics in algebraic number theory. After introducing the basic objects (algebraic integers, ideals and ideal class groups, units) we will discuss zeta functions, class number formulas, and cyclotomic fields.

Lectures:

Time: Mondays 10.30-12.30, 19 January 2009 to 16 February 2009 (The lecture on 2 February 2009 was cancelled due to the weather conditions. A replacement lecture took place on Tuesday 24 February 2009, 10.30-12.30.)
Location: De Morgan House, Russell Square
Lecturer: Dr Manuel Breuning, King's College London

Course structure and course materials:

Date	Topics	Lecture notes	Problem sheets & Solutions
19 January 2009	introduction, algebraic extensions, algebraic integers	Lecture notes, Part 1	Problem sheet 1 (Correction: In (5), one needs that A is not equal to R.) Solutions
26 January 2009	ideals, the ideal class group, units	Lecture notes, Part 1
9 February 2009	units (continued), the Riemann zeta function, the norm of an ideal	Lecture notes, Part 2	Problem sheet 2 Solutions
16 February 2009	the norm of an ideal (continued), Dedekind zeta functions, Dirichlet characters and Dirichlet L-functions	Lecture notes, Part 3
24 February 2009 (Tuesday)	cyclotomic fields, cyclotomic units	Lecture notes, Part 3

Lecture notes & exercises (Lecture notes parts 1-3 and Problem sheets 1-2 all in one file; includes some minor corrections)

Old exam questions:

Mock exam 2008
Exam 2008
Only questions (1) and (2) of the Mock exam 2008 and the Exam 2008 are relevant for this year's course, the material required for question (3) of these exams was not covered this year. Please note that this year the exam will be "take home".

Books:

The course will not follow one particular book, but there are many books which are useful as an introduction to some of the material of the course and for further reading, for example the books in the following list.

A. Fröhlich, M.J. Taylor, Algebraic number theory. CUP, 1991.
J. Neukirch, Algebraic number theory. Springer, 1999.
I. Stewart, D. Tall, Algebraic number theory and Fermat's last theorem. 3rd edition, A K Peters, 2002.
H.P.F. Swinnerton-Dyer, A brief guide to algebraic number theory. CUP, 2001.
L.C. Washington, Introduction to cyclotomic fields. Springer, 1982.

Last year's page

last modified 3 July 2009