Franz Lemmermeyer
Algebraic Number Theory
Schedule
TU 13:40 - 15:30, SAZ 03
TH 13:40 - 14:30, SAZ 20
Motivation
A standard course in algebraic number theory discusses the
proofs of the main results on integral bases, discriminants,
Dedekind rings, class groups, Dirichlet's unit theorem, etc.
In this semester, I will instead concentrate on quadratic
extensions of the rationals and of the rational function
fields and introduce elliptic curves. This will allow us to
do a lot of explicit calculations that cannot be done (by hand)
for extensions of higher degree.
Topics
- I. Quadratic Number Fields. Here we will discuss
failure of unique factorization, ideal arithmetic,
class number computation, and units. Applications:
Bachet-Mordell equations y2 = x3+k,
some simple 3-descents on elliptic curves like
x3+y3 = Az3,
possibly the cubic reciprocity law.
- II. Quadratic Function Fields. These are quadratic
extensions of the rational function fields Fp[X];
their arithmetic has a lot in common with quadratic
number fields.
- III. Arithmetic of Elliptic and Hyperelliptic Curves.
These can essentially be identified with quadratic extensions
of function fields.
Homework
Homework is always due one week after hand-out except when stated
otherwise. Solutions will be posted after all students have
turned their homework in.
Schedule
- Tu 13.09.05: Euler
and the failure of unique factorization
- Th 15.09.05: Algebraic Integers
(ring of integers; basic properties of ideals)
- Tu 20.09.05: Ideals and modules
- Th 22.09.05: NO CLASS (moved to Th. 29.09.05)
- Tu 27.09.05: Unique Factorization into prime ideals
- Th 29.09.05: Decomposition law in quadratic number fields.
- Tu 04.10.05: The Pell equation
- Th 06.10.05: Some diophantine equations
- Tu 11.10.05: Principal ideal tests, Computing units.
This is the article by Lenstra
that I recommended.
- Th 13.10.05: NO CLASS (moved to Th. 27.10)
- Tu 18.10.05: Factoring integers
- Th 20.10.05: Class groups. Here are
Chapters 1-6.
- Tu 25.10.05: Quadratic Reciprocity. Here is a
review sheet.
- Th 27.10.05: Midterm 1
(solutions)
- Tu 01.11.05: discussion of midterm
- Th 03.11.05: No class
- Tu 08.11.05: Applications to cryptography; here are
Chapters 1-6 with
an expanded Chapter 6.
- Th 10.11.05: Applications to cryptography
- Tu 15.11.05: Applications to cryptography
- Th 17.11.05: Binary Quadratic Forms.
- Tu 22.11.05: Reduction
- Th 24.11.05: No class (moved to Fr. 16.12)
- Tu 29.11.05: Here are a few practice problems
- Th 01.12.05: Reduction II
- Tu 06.12.05: Gauss Composition
- Th 08.12.05: Gauss Composition II
- Tu 13.12.05: Midterm 2; average 38.3/100.
Here are the solutions.
- Th 15.12.05: discussion of midterm, isomorphism
- Fr 16.12.05, 11:00 - 12:00; practice hour
- Mo 19.12.05, 1:30; algebraic number theory; practice for final.
- We 21.12.05, 9:30 - 11:30, SAZ04 Final.
Problems
Solutions.
Average: 53
- Final average of students who turned in homework regularly: 65
- Final average of students who turned in homework irregularly: 42
- Final average of students who did not turn in homework: 27
Do you all see what I see?
- The lecture notes
(still details missing in some proofs concerning the
group structure on classes of quadratic forms . . .)
Books
Here are a few books I recommend.
- H. Cohn, Advanced Number Theory. One of the few books with a
readable account of quadratic forms.
- D. Flath, Introduction to number theory. Probably the best
presentation of the theory of quadratic forms.
- Cox, Primes of the form x2 + ny2: Chapter 1
contains the theory of binary quadratic forms; the other two chapters
deal with class field theory and complex multiplication.
- Marcus: Number Fields. My favorite for the theory of general number
fields; it has lots of exercises.
- Ireland and Rosen: A classical introduction to modern number theory.
An all-time classic.
- Murty and Esmonde: another problem collection.
- Swinnerton-Dyer: A Brief Guide to Algebraic Number Theory:
concise, not expensive; if you want to buy a book, this is it.
There are also various lecture notes to be found
here.