This is the web page of the Mastermath course Algebraic Number Theory.

Last update: .


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.


Lectures: H.W. Lenstra
P. Stevenhagen
Problem session: Gabriel Chênevert
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.


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.


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.

Sep 9HWL/PSIntroduction: Pell's equation, Diophantine problems (ch. 1)
Sep 16HWLIdeals, definition of the Picard/class group (ch. 2)
Sep 23PSExamples, invertible = locally principal (ch. 2)
Sep 30HWLLocal properties, properness, counting norm (ch. 2+)
Oct 7PSKummer-Dedekind theorem (ch. 3)
Oct 14HWLCyclotomic rings (ch. 3)
Oct 21No class
Oct 28PSIntegral closure (ch. 3), norm & trace (ch. 4)
Nov 4HWLIntegral basis, ramification (ch. 4)
Nov 11PSExample (ch. 4), finiteness of the class group (ch. 5)
Nov 18HWLFiniteness theorems (ch. 5)
Nov 25PSComputing units and class groups (ch. 7)
Dec 2HWLGalois theory for number fields (ch. 8)
Dec 9PSThe number field sieve (see reference below)
Dec 16HWLGalois theory, continued (ch. 8)


Supplementary problems (last update: 12/01/2008)

due datehomework
Sep 16Any 5 exercises among 1.7 — 1.37
Sep 23Any 5 exercices among 1.38 — 2.28
Sep 30Any 5 new exercises in chapter 2
Oct 7Any 5 new exercises among 1.32 — 3.15
Oct 14Any 5 new exercises from chapter 3
Oct 28Any 5 new exercises from chapter 3
Nov 4Any 5 exercises from chapter 4
Nov 11Any 5 new exercises from chapter 4
Nov 18Any 5 new exercises out of: 3.40, 4.33 — 4.40, ch. 5
Nov 25Any 5 new exercises from chapter 5
Dec 27.13 + 4 other exercises from chapter 7
Dec 9Any 4 exercises from chapter 8


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.


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.


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 for the course notes used in Leiden and Delft.


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 about

Several 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.