In this course, we will study abelian extensions. Let F be a fixed finite extension of Q. Class Field Theory provides a very useful description of all finite Galois extensions K/F such that Gal(K/F) is abelian. Thus, in effect, one obtains a description of Gal(Fab/F), where Fab denotes the maximal, abelian extension of F (which is an extension of F of infinite degree). Class Field Theory also provides a complete understanding of how prime ideals of the ring of integers of F behave in the extension K/F. This description is usually referred to as the "Artin Reciprocity Law."
The roots of this subject go back to the 19th century. We will begin this course with a discussion of the famous "Law of Quadratic Reciprocity" which was first proved by Gauss at the very end of the 18th century. The statement of this theorem concerns the possible remainders that a perfect square can give when divided by a prime and seems to have nothing to do with algebraic number theory. But one of the nicest proofs of the theorem is based on the fact that every quadratic extension of Q is a subfield of some cyclotomic extension of Q. This result is a special case of the much more difficult "Kronecker-Weber" theorem which asserts that if K is any finite, abelian extension of Q, then K is a subfield of some cyclotomic extension of Q. To state this another way, Qab = Q(W), where W denotes the set of all roots of unity. It is natural to ask whether there is a "Law of Cubic Reciprocity." During the 19th century, this question and its natural generalizations occupied the attention of Gauss and many other number theorists. Such a law does indeed exist and the modern point of view is to formulate such a law in the context of Class Field Theory. Then it becomes a special case of the Artin Reciprocity Law.
The prerequisites for this course include a thorough understanding of the basic theory of commutative rings and fields and of Galois Theory. Familiarity with algebraic number theory would probably be helpful, but the main definitions and results will be reviewed. The course will begin with the Quadratic Reciprocity Law, which we will prove in several ways. This leads naturally to the study of quadratic fields, which in turn leads to the study of cyclotomic fields. We will prove the Kronecker-Weber Theorem. This will lead to the statements of the main theorems of Class Field Theory. It will not be possible to prove these theorems completely in this course, but we will discuss a variety of illustrations of how they can be applied, including in particular a Cubic Reciprocity Law. Here is one specific application which seems on the surface to have nothing to do with algebraic number theory: Suppose that p is a prime of the form 3k+1. Then there exists a perfect cube which gives a remainder of 2 when divided by p if and only if p can be expressed in the form x2 + 27y2, where x and y are integers.
There will be no required textbook for the course. Several books will be suggested for background reading.