This is the web page of the Mastermath course Algebraic Number Theory.
Last update: .
Announcements
If you want to get all your graded homeworks back, as well as your final grade, at the last lecture (Dec. 16) you need to submit all remaining assignments by December 9.
After this, it will still be possible to submit late homeworks (or additional exercises!), but no later than Friday, January 16.
Organization
| Lectures: | H.W. Lenstra | hwl@math.leidenuniv.nl |
| P. Stevenhagen | psh@math.leidenuniv.nl | |
| Problem session: | Gabriel Chênevert | gcheneve@math.leidenuniv.nl |
| Location: | W&N Building S205, Vrije Universiteit, Amsterdam |
| Time: | Tuesdays, 10:15 – 13:00 (2 hours of lecture, 1 hour problem session) First lecture: September 9, 2008 Last lecture: December 16, 2008 No lecture on Tuesday, October 21. |
Aim
The course provides a thorough introduction to algebraic number theory. It treats the basic laws of arithmetic that are valid in subrings of algebraic number fields.
Program
As a general rule, the lectures will follow the course notes. To make the best use of class time, you might want to take a look at the appropriate sections in advance.
| date | lecturer | subject |
| Sep 9 | HWL/PS | Introduction: Pell's equation, Diophantine problems (ch. 1) |
| Sep 16 | HWL | Ideals, definition of the Picard/class group (ch. 2) |
| Sep 23 | PS | Examples, invertible = locally principal (ch. 2) |
| Sep 30 | HWL | Local properties, properness, counting norm (ch. 2+) |
| Oct 7 | PS | Kummer-Dedekind theorem (ch. 3) |
| Oct 14 | HWL | Cyclotomic rings (ch. 3) |
| Oct 21 | — | No class |
| Oct 28 | PS | Integral closure (ch. 3), norm & trace (ch. 4) |
| Nov 4 | HWL | Integral basis, ramification (ch. 4) |
| Nov 11 | PS | Example (ch. 4), finiteness of the class group (ch. 5) |
| Nov 18 | HWL | Finiteness theorems (ch. 5) |
| Nov 25 | PS | Computing units and class groups (ch. 7) |
| Dec 2 | HWL | Galois theory for number fields (ch. 8) |
| Dec 9 | PS | The number field sieve (see reference below) |
| Dec 16 | HWL | Galois theory, continued (ch. 8) |
Homework
Supplementary problems (last update: 12/01/2008)
| due date | homework |
| Sep 16 | Any 5 exercises among 1.7 — 1.37 |
| Sep 23 | Any 5 exercices among 1.38 — 2.28 |
| Sep 30 | Any 5 new exercises in chapter 2 |
| Oct 7 | Any 5 new exercises among 1.32 — 3.15 |
| Oct 14 | Any 5 new exercises from chapter 3 |
| Oct 28 | Any 5 new exercises from chapter 3 |
| Nov 4 | Any 5 exercises from chapter 4 |
| Nov 11 | Any 5 new exercises from chapter 4 |
| Nov 18 | Any 5 new exercises out of: 3.40, 4.33 — 4.40, ch. 5 |
| Nov 25 | Any 5 new exercises from chapter 5 |
| Dec 2 | 7.13 + 4 other exercises from chapter 7 |
| Dec 9 | Any 4 exercises from chapter 8 |
Description
Introduction to algebraic numbers and number rings. Ideal factorization, finiteness results on class groups and units, explicit computation of these invariants. Possible special topics: binary quadratic forms, the number field sieve, valuations and completions, local fields, introduction to class field theory and reciprocity laws, density theorems.
Examination
The final grade is exclusively based on the results obtained for the weekly homework assignments. Collaboration on these problems is allowed, but every student should submit their own redaction: the grade of a copy will be divided by the number of copies in its isomorphism class. In order to learn something from the class, you should try to solve and submit problems that you find challenging at first sight.
Prerequisites
Undergraduate algebra, i.e., the basic properties of groups, rings, and fields, including Galois theory. This material is covered in first and second year algebra courses in the bachelor program of most universities. See http://www.math.leidenuniv.nl/algebra for the course notes used in Leiden and Delft.
Literature
In 2004 and 2006, the course was taught using the course notes Number rings on the Leiden algebra page.
A survey paper following roughly the same line of exposition is Stevenhagen's
in the Algorithmic Number Theory-volume appearing at Cambridge University Press. The volume has an introductory paper by Lenstra entitled as well as another survey by Stevenhagen aboutSeveral books entitled `Algebraic number theory', such as those by Stewart & Tall, E. Weiss, S. Lang, J. Neukirch, or Cassels & Fröhlich, can also profitably be consulted.