Math 581CA-582CA
Algebraic Number Theory

Ralph Greenberg

Autumn/Winter 2002-2003, Monday/Wednesday/Friday 10:30-11:20

This course will be a traditional introduction to algebraic number theory. If K is a finite extension of Q, then there is a natural subring of K, namely the ring R of algebraic integers in K. That is, R is the integral closure of Z in K. The ring R will be the main object of study in this course. Here are the main topics:

1. The ideal theory of R. The unique factorization theorem does not hold in the traditional sense. However, there is a useful analogue for ideals. The main theorem, which will be proved rather early in the course, states that every nonzero ideal of R can be expressed uniquely as a product of prime ideals of R.

2. The ideal class group of R. One puts an equivalence relation on the nonzero ideals of R: Two nonzero ideals I and J of R are "equivalent" if and only if there exists nonzero principal ideals A and B of R such that IA = JB. The equivalence classes turn out to form an abelian group. The main theorem about this topic is that this group is finite. The ideal class group has been an object of extensive study since the 19th century. We will have a lot to say about it in this course.

3. Units. The group of units, or invertible elements, of the ring R is, of course, an important topic of study. The main theorem that we will prove about it is "Dirichlet's Unit Theorem." This describes the structure of the group of units as an abstract group. The proof brings another branch of number theory into the subject - the geometry of numbers.

4. Decomposition of primes. This is one of the most important aspects of algebraic number theory. Let p be a prime number. The principal ideal of R generated by p factors as a product of nonzero prime ideals (as mentioned in (1).) How does this factorization depend on p? In general, this is essentially an unsolved question. But in certain special cases, one can give a complete answer. One consequence is that one gets a very natural proof of a theorem in classical number theory - the Law of Quadratic Reciprocity.

Other topics that will be studied in the course are: Completions, Zeta Functions, and Class Field Theory, to the extent that time permits.

Prerequisites: A good knowledge of basic ring theory and group theory is essential. Also, Galois theory will play an important role. Certainly, the first year algebra course should provide sufficient background.

Text: Number Fields, Daniel Marcus