This is the web page for the Dutch national master course algebraic number theory, fall 2006. For information on other national master courses, visit Mastermath.

Announcements

Organization

Lectures by: Bart de Smit (Leiden) and Jaap Top (Groningen).
Teaching assistants: Stephen Meagher (Groningen) and Willem Jan Palenstijn (Leiden).
Time: Mondays from September 11 - December 18, 2006, 10:15 - 13:00.
Location: Room S209 of the Mathematics and Sciences building (WG, 1081), Vrije Universiteit, Amsterdam. Directions.

The final hour (12:15-13:00) will be devoted to homework problems. Successfully completing this course will be rewarded with 8 EC points.

Homework

Each week, students have to hand in 4 exercises from the course notes out of those listed below. Solving the more difficult problems will result in a higher grade. The final problem set of the course will be more substantial. Note that the final grade for this course is based exclusively on the results obtained for the weekly homework assignments.

The regular exercises are worth 2 points each, and the challenging exercises 3 points. The grade for each week is computed by multiplying the total number of points by 9/12, and adding 1.

When handing in homework, ensure that each exercise is on a separate sheet of paper. English and Dutch are both accepted.

The teaching assistants are always available by e-mail or in person if you require assistance.

Due date Exercises Challenging exercises
Monday September 18 §1: 8, 10, 16, 18, 22 §1: 9, 11, 12, 17, 25, 30
Monday September 25 §2: 17, 18, 34, 35 §2: 23, 25, 39, 40, 45
Monday October 2 §2: 13, 24, 29, 31 §2: 19, 27, 32, 36, 46
Monday October 9 §3: 4, 5, 9, 12, 29 §3: 15, 18, 19, 20
Monday October 16 §3: 3, 8, 10, 11, 13 §3: 12[1], 14, 16, 17
Monday October 30 §3: 6, 7, 21, 22, 24 §3: 26, 27, 29, 31
Monday November 6 §4: 6, 9, 12, 15, 18 §4: 5, 10, 13, 16
Monday November 13 §4: 3, 23, 24, 27 §4: 11, 25, 26, 28, 29
Monday November 20 §5: 7, 9, 10, 12 §5: 4, 11, 17, §3.28+§5.1[2][3]
Monday November 27 §5: 20, 23, 29, extra[4] §5: 21, 22, 30, 33
Monday December 4 §7: 5, 6, 7, 8, 9 §7: 12, 13, 16, 17
Monday January 15 Take-home exam

Remarks

  1. Replace 7 by 9 in exercise 3.12 for October 16.
  2. Exercises 3.28 and 5.1 together count as one challenging exercise for November 20.
  3. There is an error in exercise 3.28. The direct sum in the center of the exact sequence should be over p in S.
  4. In the extra exercise, m = (p + 1)/4.

Description

The course provides a thorough introduction to algebraic number theory. It treats the arithmetic of the number rings that occur in (algorithmic) practice.

Topics: Introduction to algebraic numbers and number rings. Ideal factorization, finiteness results on class groups and units, explicit computation of these invariants. Special topic: the number field sieve.

Prerequisites: Undergraduate algebra, i.e., the basic properties of groups, rings and fields. This material is covered in first and second year algebra courses in the bachelor program of most universities. The course notes used in Leiden (in Dutch) and those used in Groningen (also in Dutch) are available online.

Literature

We will use the course notes and homework exercises of Peter Stevenhagen. Further recommended books: I.N. Stewart & D.A. Tall, Algebraic number theory; P. Samuel, Algebraic theory of numbers; D.A. Marcus, Number Fields

Examination

The final grade is exclusively based on the results obtained for the weekly homework assignments. The last problem set will be more substantial and determine one third of the final grade.