| Instructor: |
Nils Bruin
nbruin@sfu.ca
SC K 10507
(778) 782 3794
|
| Webpage: |
http://www.cecm.sfu.ca/~nbruin/math842/ |
| Outline: |
Algebraic number theory comprises the study of algebraic numbers:
numbers that satisfy polynomial equations with rational coefficients.
The parallels with usual integer arithmetic are striking, as are the
notable differences (as, for instance, failure of unique factorization
into prime factors). The subject is fundamental to any further study in
number theory or algebraic geometry.
In this course we develop the tools to properly understand unique
factorization and its failure. We establish fundamental results such as
Dirichlet's Unit theorem and the finiteness of the ideal class group.
We highlight the applicability of the algebraic tools we develop to
both algebraic numbers and to algebraic curves. Depending on time and
interests of the participants, we will also look into various
applications and more advanced topics.
Possible topics:
- Dedekind domains: ideals, fractional ideals, ideal factorization, and ideal class groups
- Lattices and Minkowski Theory.
- Dirichlet's Unit theorem
- Finiteness of class groups
- Ramification theory; interaction with Galois groups
- Cyclotomic fields
- Completions and valuations
- Adelic approaches
- Absolute Galois groups; profinite groups
- Group cohomology; Galois cohomology
|
| Course text: |
Milne, J.S.,
Algebraic Number Theory.
available from:
http://www.jmilne.org/math/CourseNotes/ant.html.
(local copy) |
Recommended reading: |
- Quite algebraic and very careful exposition, including lots of detail in proofs:
Ribenboim, Paulo,
Classical theory of algebraic numbers.
Universitext. Springer-Verlag, New York, 2001. xxiv+681 pp. ISBN: 0-387-95070-2
- Very compact in its exposition, but still complete. Probably one of the
slickest presentations available:
Neukirch, Jürgen,
Algebraic number theory.
Grundlehren der Mathematischen Wissenschaften 322.
Springer-Verlag, Berlin, 1999. xviii+571 pp. ISBN: 3-540-65399-6
- Quite complete in its coverage of theory but also paying attention to
the computational side of things:
Stein, William
Algebraic Number Theory, a Computational Approach.
available from: https://wstein.org/books/ant/.
(local copy)
|
| Lectures: |
Wednesdays, Fridays 14:30 - 16:20 in WMC 2501
Some Mondays (in Bold) 15:30 - 17:20 in WMC 2830
First lecture: Wednesday January 15
|
| Exam: |
April 14, 9:00 - 17:00 (take-home final)
Accommodation will be made for clashes with Prel/Comp exams |
| Grading: |
| Preliminary scheme: |
| Assignments: |
30 % |
| Presentation: |
15 % |
| Exam: |
55 % |
|
| Lecture schedule: |
PRELIMINARY; we may not need all booked Mondays.
References to Milne are in terms of Definition/Remark/Theorem etc. numbers.
| Date |
Subject |
References |
| Jan. 15 |
Cancelled due to snow |
|
| Jan. 17 |
Introduction: rings of integers |
[Milne 2.1-3] |
| Jan. 20 |
First properties of integral closures: Dedekind's characterization of integrality via modules; UFDs are integrally closed |
[Milne 2.4-14] |
| Jan. 22 |
Trace form; Discriminants |
[Milne 2.15-2.23] |
| Jan. 24 |
Finite generation of integer rings; how to compute them |
[Milne 2.24-43], we did some extra things too. For Assignment 0:Convergent Calculator |
| Jan. 27 |
Discrete valuation rings and localization |
[Milne 3.1-3.2; 1.10-1.13] |
| Jan. 29 |
Unique ideal factorization in Dedekind domains |
[Milne 3.3-3.18] |
| Jan. 31 |
The ideal class group |
[Milne 3.19-3.30] |
| Feb. 5 |
Ideal factorization in extensions |
[Milne 3.33-3.34] |
| Feb. 7 |
Ramification |
[Milne 3.35-3.53 |
| Feb. 12 |
No Lecture |
| Feb. 14 |
No Lecture |
| Reading break |
| Feb. 24 |
Norms of ideals; Minkowski bound |
[Milne 4.1-4.10 |
| Feb. 26 |
Lattices |
[Milne 4.11-16], proof that a discrete subgroup of $\mathbb{R}^n$ is a lattice |
| Feb. 28 |
Minkowski's convex body theorem: The class group is finite |
[Milne 4.17-4.28] |
| Mar. 2 |
The unit group: finite generation |
[Milne 5.1-5.8] |
| Mar. 4 |
The unit rank, S-units, and computing class groups and unit groups. |
[Milne 5.9-5.11] |
| Mar. 6 |
No Lecture |
| Mar. 11 |
Ostrowski's Theorem |
[Milne 7.1-7.12] |
| Mar. 13 |
Fermat's theorem for regular primes (first case only) |
[Milne 6.8-6.11, plus earlier results on cyclotomic extensions |
| Mar. 18 |
Weak approximation |
For assignment 3: Sage code for computing valuationsi
[Milne 7.1-7.21] |
| Mar. 20 |
Completions |
[Milne 7.23-7.30] |
| Mar. 23 |
Hensel's Lemma; norms on vector spaces over complete fields |
[Milne 7.31-7.37] (Milne does not do the norms on vector spaces in sufficient generality) |
| Mar. 25 |
Newton polygons; extensions |
[7.38-7.46] (We didn't quite follow this approach) |
| Mar. 27 |
Ramified and unramified extensions; Krasner's Lemma |
[Milne 7.47-7.65] |
| Mar. 30 |
Global fields; Strong approximation; Decomposition groups |
[Milne 8.10-8.11, 8.18-8.19] -- strong approximation is not in Milne |
| Apr. 1 |
Kummer theory: Cohomology, Galois modules, Hilbert 90, and the absolute Galois group |
Not in Milne |
| Apr. 3 |
Presentations |
| Apr. 8 |
Presentations |
| Apr. 14 |
Final Exam 9:00 - 17:00 |
|