Instructors:
The lectures will be given on Thursday morning from 10:15 until 13:00 in
room P.016 of the Euclides building (Plantage Muidergracht 24) of the
University of Amsterdam.
The final hour (12:15-13:00) will be devoted to homework problems.
The lectures start on September 9th and continue until December 16th; there
won't be any lecture on October 14th.
Attending this course will be rewarded with 8 EC points.
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-Delft can be found here.
There will be course notes and homework exercises (see below). The book `Algebraic number theory and Fermat's last theorem' by Stewart and Tall and several books entitled `Algebraic number theory', such as those by E. Weis, S. Lang, J. Neukirch or Cassels and Fröhlich, can be profitably consulted. Unfortunately, the book by Cassels and Fröhlich is out of print and the cheapest second hand copy is $177. If you find a cheaper one, please let us know! The book of Weis is also out of print, but a lot of cheap (less than $10) copies are available via AddAll.
Introduction to algebraic numbers and number rings. Ideal factorization, finiteness results on class groups and units, explicit computation of these invariants. Special topics: binary quadratic forms, the number field sieve. Valuations and completions, local fields, introduction to class field theory. Special topics: reciprocity laws, representation of primes by quadratic forms, density theorems.
The final grade is exclusively based on the results obtained for the weekly homework assignments. The final homework assignment will be slightly more substantial.
Notes (ps);
Notes (pdf);
Sieving methods (ps),
Sieving methods (pdf);
Number field sieve (ps),
Number field sieve (pdf);
Chebotarev;
Artin.
The two main software tools for computations in algebraic number fields
are Pari and
Magma. Pari has the big
advantage that it is free, but it is not easy to use as Magma. (This is of
course my own personal opinion.) Unfortunately, Magma is not free. However,
an online `calculator' for both packages is available
here.
I have put three simple Magma scripts online. The script
euler can be
used to compute Euler product for polynomials of degree 4. The script
nfex is a
simple script showing some Magma commands relating algebraic number fiels.
Finally, the script
gcdex shows
the correct way to compute the gcd of x^p-x and an other polynomial f when
p is large. (This went a bit wrong in class...)
Everyone has to do 4 exercises a week. Below is the subset from which you may choose them. A subset of the `difficult' exercises is also given. Note that some `explicit computation' exercises are marked difficult. This is not because they really are, but because I want to encourage you to do them!
| Date | Exercises | Difficult ones | ||
| 16-09-2004 | § 1: 1-31 | |||
| 23-09-2004 | § 2: 1-56 | |||
| 30-09-2004 | § 2: 1-56, § 3:1-31 | § 2: 13,15,16,18,23,26,28,34,41,42,45,46; § 3: 2 from 10-12, 16,18,20,25,29,30,31 | ||
| 07-10-2004 | § 3:1-31 | § 3: 2 from 10-12, 16,18,20,25,29,30,31 | ||
| 21-10-2004 | § 4:1-32 | § 4: 7,10,13,14,16,17,20,24,26,29,31 | ||
| 28-10-2004 | § 4:1-32 | § 4: 7,10,13,14,16,17,20,24,25,26,27,28,29,31 | ||
| 04-11-2004 | § 5:1,2,4-20 | § 5: 4,6,8,9,12,13,17,19 | ||
| 11-11-2004 | § 5:1-35 | § 5: 21,24,25,31,32,34,35 | ||
| 18-11-2004 | class group exercise (ps), class group exercise (pdf) | |||
| 25-11-2004 | § 7:1-17 | § 7: 5-17 | ||
| 02-12-2004 | catch up week | |||
| 09-12-2004 | no homework | |||
| 16-12-2004 | final exercises (ps), final exercises (pdf) |
Note: everyone has to do problem zero from the `final exercises'. As usual, it suffices to do four exercises.