Kenneth Ireland

The Zeta Function

The zeta function of an algebraic variety has played a major role in recent developments in diophantine geometry.

Quadratic Gauss Sums

The method by which we proved the quadratic reciprocity in Chapter 5 is ingenious but is not easy to use in more general situations. We shall give a new proof in this chapter that is based on methods that can be used to prove higher reciprocity laws. In particular, we shall introduce the notion of a Gauss sum, which will play an important role in the latter part of this book.

New Progress in Arithmetic Geometry

The decade of the eighties saw dramatic progress in the field of arithmetic geometry. Problems that were previously thought to be inaccessible by contemporary methods were in fact resolved. It is the purpose of this chapter to survey a portion of these dramatic developments.

Algebraic Number Theory

In this chapter we shall introduce the concept of an algebraic number field and develop its basic properties. Our treatment will be classical, developing directly only those aspects that will be needed in subsequent chapters. The study of these fields, and their interaction with other branches of mathematics forms a vast area of current research. Our objective is to develop as much of the general theory as is needed to study higher-power reciprocity. The reader who is interested in a more systematic treatment of these fields should consult any one of the standard texts on this subject, e.g., Ribenboim [207], Lang [168], Goldstein [140], Marcus [183].

Equations over Finite Fields

In this chapter we shall introduce a new point of view. Diophantine problems over finite fields will be put into the context of elementary algebraic geometry. The notions of affine space, projective space, and points at infinity will be defined.

The Mordell-Weil Theorem

In this chapter we prove the celebrated theorem of Mordell—Weil for elliptic curves defined over the field of rational numbers. Our treatment is elementary in the sense that no sophisticated results from algebraic geometry are assumed. It is our desire to present a self-contained treatment of this important result. The significance and implications of this theorem for contemporary research in diophantine geometry are farreaching. In the following chapter a summary without proofs of these developments to the present time is sketched. We hope that these two chapters will inspire the interested student to continue this study by consulting the more comprehensive texts on the arithmetic of elliptic curves listed in the bibliography to this chapter.

The Structure of U (ℤ/ n ℤ)

Having introduced the notion of congruence and discussed some of its properties and applications we shall now go more deeply into the subject. The key result is the existence of primitive roots modulo a prime. This theorem was used by mathematicians before Gauss but he was the first to give a proof. In the terminology introduced in Chapter 3 the existence of primitive roots is equivalent to the fact that U(ℤ/nℤ) is a cyclic group when p is a prime. Using this fact we shall find an explicit description of the group U(ℤ/nℤ) for arbitrary n.

Applications of Unique Factorization

The importance of the notion of prime number should be evident from the results of Chapter 1.

The Stickelberger Relation and the Eisenstein Reciprocity Law

Having developed the basic properties of cyclotomic fields we will prove two beautiful and important theorems which play a fundamental role in the further development of the theory of these fields.