A Classical Introduction to Modern Number Theory

Frontmatter

Chapter 1. Unique Factorization

Abstract

The notion of prime number is fundamental in number theory. The first part of this chapter is devoted to proving that every integer can be written as a product of primes in an essentially unique way.

After that, we shall prove an analogous theorem in the ring of polynomials over a field.

On a more abstract plane, the general idea of unique factorization is treated for principal ideal domains.

Finally, returning from the abstract to the concrete, the general theory is applied to two special rings that will be important later in the book.

Kenneth Ireland, Michael Rosen

Chapter 2. Applications of Unique Factorization

Abstract

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

In this chapter we shall give several proofs of the fact that there are infinitely many primes in ℤ. We shall also consider the analogous question for the ring k[x].

The theorem of unique prime decomposition is sometimes referred to as the fundamental theorem of arithmetic. We shall begin to demonstrate its usefulness by using it to investigate the properties of some natural number-theoretic functions.

Kenneth Ireland, Michael Rosen

Chapter 3. Congruence

Abstract

Gauss first introduced the notion of congruence in Disquisitiones Arithmeticae (see Notes in Chapter 1). It is an extremely simple idea. Nevertheless, its importance and usefulness in number theory cannot be exaggerated.

This chapter is devoted to an exposition of the simplest properties of congruence. In Chapter 4, we shall go into the subject in more depth.

Kenneth Ireland, Michael Rosen

Chapter 4. The Structure of U(ℤ/nℤ)

Abstract

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/pℤ) 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.

Kenneth Ireland, Michael Rosen

Chapter 5. Quadratic Reciprocity

Abstract

If p is a prime, the discussion of the congruence x ²≡ a(p) is fairly easy. It is solvable iff a ^(p-1)/2≡ 1 (p). With this fact in hand a complete analysis is a simple matter. However, if the question is turned around, the problem is much more difficult. Suppose that a is an integer. For which primes p is the congruence x ²≡ a(p) solvable? The answer is provided by the law of quadratic reciprocity. This law was formulated by Euler and A. M. Legendre but Gauss was the first to provide a complete proof. Gauss was extremely proud of this result. He called it the Theorema Aureum, the golden theorem.

Kenneth Ireland, Michael Rosen

Chapter 6. Quadratic Gauss Sums

Abstract

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.

Section 1 introduces algebraic numbers and algebraic integers. The proofs are somewhat technical. The reader may wish to simply skim this section on a first reading.

Kenneth Ireland, Michael Rosen

Chapter 7. Finite Fields

Abstract

We have already met with examples of finite fields, namely, the fields ℤ/pℤ, where p is a prime number. In this chapter we shall prove that there are many more finite fields and shall investigate their properties. This theory is beautiful and interesting in itself and, moreover, is a very useful tool in number-theoretic investigations. As an illustration of the latter point, we shall supply yet another proof of the law of quadratic reciprocity. Other applications will come later.

One more comment. Up to now the great majority of our proofs have used very few results from abstract algebra. Although nowhere in this book will we use very sophisticated results from algebra, from now on we shall assume that the reader has some familiarity with the material in a standard undergraduate course in the subject.

Kenneth Ireland, Michael Rosen

Chapter 8. Gauss and Jacobi Sums

Abstract

In Chapter 6 we introduced the notion of a quadratic Gauss sum. In this chapter a more general notion of Gauss sum will be introduced. These sums have many applications. They will be used in Chapter 9 as a tool in the proofs of the laws of cubic and biquadratic reciprocity. Here we shall consider the problem of counting the number of solutions of equations with coefficients in a finite field. In this connection, the notion of a Jacobi sum arises in a natural way. Jacobi sums are interesting in their own right, and we shall develop some of their properties.

To keep matters as simple as possible, we shall confine our attention to the finite field ℤ/pℤ = F _pand come back later to the question of associating Gauss sums with an arbitrary finite field.

Kenneth Ireland, Michael Rosen

Chapter 9. Cubic and Biquadratic Reciprocity

Abstract

In Chapter 5 we saw that the law of quadratic reciprocity provided the answer to the question. For which primes p is the congruence x ²≡ a (p) solvable ? Here a is a fixed integer. If the same question is considered for congruences x ⁿ≡ a (p), n a fixed positive integer, we are led into the realm of the higher reciprocity laws. When n = 3 and 4 we speak of cubic and biquadratic reciprocity.

Kenneth Ireland, Michael Rosen

Chapter 10. Equations over Finite Fields

Abstract

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.

Kenneth Ireland, Michael Rosen

Chapter 11. The Zeta Function

Abstract

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

Kenneth Ireland, Michael Rosen

Chapter 12. Algebraic Number Theory

Abstract

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].

Kenneth Ireland, Michael Rosen

Chapter 13. Quadratic and Cyclotomic Fields

Abstract

In the last chapter we discussed the general theory of algebraic number fields and their rings of integers. We now consider in greater detail two important classes of these fields which were studied first in the nineteenth century by Gauss, Eisenstein, Kummer, Dirichlet, and others in connection with the theory of quadratic forms, higher reciprocity laws and Fermâtes Last Theorem. The reader who is interested in the historical development of this subject should consult the book by H. Edwards [128] as well as the classical treatise by H. Smith [72].

Kenneth Ireland, Michael Rosen

Chapter 14. The Stickelberger Relation and the Eisenstein Reciprocity Law

Abstract

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.

Kenneth Ireland, Michael Rosen

Chapter 15. Bernoulli Numbers

Abstract

In this chapter we will introduce an important sequence of rational numbers discovered by Jacob Bernoulli (1654–1705) and discussed by him in a posthumous work Ars Conjectandi (1713). These numbers, now called Bernoulli numbers, appear in many different areas of mathematics. In the first section we give their definition and discuss their connection with three different classical problems. In the next section we discuss various arithmetical properties of Bernoulli numbers including the Claussen-von Staudt theorem and the Kummer congruences. The first of these results determines the denominators of the Bernoulli numbers, and the second gives information about their numerators. In the last section we prove a theorem due to J. Herbrand which relates Bernoulli numbers to the structure of the ideal class group of ℚ(ζ _p). The material in this section is somewhat sophisticated but we have included it anyway because it provides a beautiful and important application of the Stickelberger relation which was proven in the last chapter.

Kenneth Ireland, Michael Rosen

Chapter 16. Dirichlet L-functions

Abstract

The theory of analytic functions has many applications in number theory. A particularly spectacular application was discovered by Dirichlet who proved in 1837 that there are infinitely many primes in any arithmetic progression b, b + m, b + 2m, … , where (m, b) = 1. To do this he introduced the L-functions which bear his name. In this chapter we w ill defi n e these functions, investigate their properties, and prove the theorem on arithmetic progressions. The use of Dirichlet L-functions extends beyond the proof of this theorem. It turns out that their values at negative integers are especially important. We will derive these values and show how they relate to Bernoulli numbers.

Kenneth Ireland, Michael Rosen

Chapter 17. Diophantine Equations

Abstract

In Chapter10 we discussed Diophantine equations over finitefields. In this chapter we consider special Diophantine equations with integral coefficients and seek integral or rational solutions. The techniques used vary from elementary congruence considerations to the use of more sophisticated results in algebraic number theory. In addition to establishing the existence or nonexistence of solutions we also obtain results of a quantitative nature, as in the determination of the number of representations of an integer as the sum of four squares. All of the equations considered in this chapter are classical, each playing an important role in the historical development of the subject.

Kenneth Ireland, Michael Rosen

Chapter 18. Elliptic Curves

Abstract

Many of the themes studied throughout this book corme together in the arithmetic theory of elliptic curves. This is a branch of number theory whose roots go back a long way, but which is, nevertheless, the subject of intense investigation at the present time.

Kenneth Ireland, Michael Rosen

Chapter 19. The Mordell-Weil Theorem

Abstract

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.

Kenneth Ireland, Michael Rosen

Chapter 20. New Progress in Arithmetic Geometry

Abstract

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.

Kenneth Ireland, Michael Rosen

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