Algebraic Number Theory
Mo 13:40 - 15:30, SAZ 02
We 15:40 - 17:30, SAZ 02
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.
- 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
- III. Arithmetic of Elliptic and Hyperelliptic Curves.
These can essentially be identified with quadratic extensions
of function fields.
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.
- Mo 11.09.06: Euler and the failure of unique factorization
- We 13.09.06: Euclidean rings; Principal Ideal Domains
- Mo 18.09.06: Unique Factorization Domains; Ideals
- We 20.09.06: Ideals
- Mo 25.09.06: Algebraic Integers, modules
- We 27.09.06, 4:40 - 5:30 pm Ideals.
- Mo 02.10.06: Unique Factorization into prime ideals
- We 04.10.06 3:40 - 4:30 examples
- Mo 09.10.06: Units
- We 11.10.06 3:40 - 5:30: Units
- Mo 16.10.06 Class groups
- We 18.10.06: NO CLASS, moved to 11.10.06
- Mo 23.10.06 no class
- We 25.10.06 no class
- Mo 30.10.06 Class group calculations continued
- We 02.11.06 Diophantine equations
- Mo 06.11.06 Genus theory and quadratic reciprocity
- We 09.11.06 3:40 - 5:30: quadratic forms
- Mo 13.11.06 quadratic forms
- We 15.11.06 no class; moved to 09.11.
- Sa 18.11.06 12:00 - 14:00, SAZ04: Midterm 1 covering everything
up to class group calculations. Proctor: Adem Ersin Ureyen.
Here are the solutions
- Mo 20.11.06 reduction of quadratic forms
- We 22.11.06 quadratic forms
- Mo 27.11.06 Bhargava composition of quadratic forms
- We 29.11.06 No Class.
- Mo 04.12.06 Class groups in orders
- We 06.12.06 Bijection between forms and modules.
- Mo 11.12.06 Quadratic Forms in Function Fields
- We 13.12.06 Elliptic and Hyperelliptic Curves.
Here are the updated
- Sa 16.12.06 15:00, SAZ 04, Midterm 2.
Here are the solutions.
- Mo 18.12.06 Elliptic and Hyperelliptic Curves
- We 20.12.06 No class (16.12.)
- Jan 04, 2007 NO FINAL (postponed)
- Jan 06, 2007; 14:00 - 16:00, SAZ03 final.
Here are a few books I recommend.
There are also various lecture notes to be found
- 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.