One of the most interesting objects of study in algebraic number theory is the ideal class group of a number field. If this group is trivial, then the ring of integers of the number field is a PID. For the special case of quadratic number fields, the ideal class groups can be computed and one finds that their structure and order are rather unpredictable. In the late 1950's, Iwasawa began to study certain towers of number fields and discovered that their ideal class groups exhibited some regularity in their structure and order. This discovery has turned out to be extremely fruitful. It eventually led to a deep understanding of relationships between the structure of ideal class groups of number fields and certain values of functions like the Riemann zeta function. (In the case of the Riemann zeta function, the first hint of a relationship with ideal class groups occurred in the work of Kummer on Fermat's Last Theorem.) Iwasawa's discovery has also developed into one of the most powerful ways of studying questions about the arithmetic of elliptic curves.

In this course we will go back to the beginning of the subject. The towers of number fields which Iwasawa studied are defined in terms of their Galois groups. Let K be any number field. Let p be a fixed prime. Consider a tower of number fields K K₁ K₂ ... K_n ... , where K_n/K is a cyclic extension of degree pⁿ for all positive n. (This means that K_n/K is Galois and Gal(K_n/K) is a cyclic group of order pⁿ.) The first important theorem that we will prove is that the order of the p-primary subgroup of the ideal class group of K_n is given by a simple formula which depends on two invariants (lambda and mu). We will prove various theorems about these invariants. One of the tools in studying this subject is the structure theory of modules over the formal power series ring Z_p[[T]], where Z_p is the ring of p-adic integers. Although this ring is not a PID, one can say enough about the module-theory of this ring to effectively study the ideal class groups of the K_n's. (We will leave in suspense right now the beautiful way in which the module-theory of Z_p[[T]] can be linked to the ideal class groups.)

There will be no textbook for this course. Several helpful books will be put on reserve. The prerequisites for this course are a thorough understanding of Galois theory (including Galois theory for infinite extensions), a good understanding of basic algebraic number theory (including ideal class groups, Dirichlet's unit theorem, and completions of number fields), and some commutative ring theory. Although class field theory will play an important role in this course, we will explain the basic theorems of that subject.