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
- The results of the take home exam are now available.
- The deadline for the take home exam is 15 January 2007. This is also the final deadline for any homework you have not yet handed in. Please fill out the evaluation form attached to the exam to help us improve this course the next time it will be taught. (N.B., all exercises of the exam are worth the same number of points.)
- There was an error in exercise 7 of the exam. The equation should be a^{p} + b^{p} = c^{p}.
- If you want to hand in the take home exam by mail, the address to mail it to is:
W.J. Palenstijn
Mathematisch Instituut, Universiteit Leiden
Postbus 9512
2300 RA Leiden
- Don't forget to register for this course.
- The example Magma code from class is available here.
- Extra texts on factoring: Sieving Methods by Carl Pomerance, and the Number Field Sieve by Peter Stevenhagen.
- In spring 2007, there is a Topics in number theory course in Leiden, which builds on the material covered in the present course.
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
- Replace 7 by 9 in exercise 3.12 for October 16.
- Exercises 3.28 and 5.1 together count as one challenging exercise for November 20.
- There is an error in exercise 3.28. The direct sum in the center of the exact sequence should be over p in S.
- 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.