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1982 | Buch

A Classical Introduction to Modern Number Theory

verfasst von: Kenneth Ireland, Michael Rosen

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

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Über dieses Buch

This book is a revised and greatly expanded version of our book Elements of Number Theory published in 1972. As with the first book the primary audience we envisage consists of upper level undergraduate mathematics majors and graduate students. We have assumed some familiarity with the material in a standard undergraduate course in abstract algebra. A large portion of Chapters 1-11 can be read even without such background with the aid of a small amount of supplementary reading. The later chapters assume some knowledge of Galois theory, and in Chapters 16 and 18 an acquaintance with the theory of complex variables is necessary. Number theory is an ancient subject and its content is vast. Any intro­ ductory book must, of necessity, make a very limited selection from the fascinat ing array of possible topics. Our focus is on topics which point in the direction of algebraic number theory and arithmetic algebraic geometry. By a careful selection of subject matter we have found it possible to exposit some rather advanced material without requiring very much in the way oftechnical background. Most of this material is classical in the sense that is was dis­ covered during the nineteenth century and earlier, but it is also modern because it is intimately related to important research going on at the present time.

Inhaltsverzeichnis

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.
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.
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.
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/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.
Kenneth Ireland, Michael Rosen
Chapter 5. Quadratic Reciprocity
Abstract
If p is a prime, the discussion of the congruence x2 ≡ a (p) is fairly easy. It is solvable iff a(p − 1)/2 ≡ (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 x2 ≡ 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.
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.
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.
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 x2 ≡ a (p) solvable? Here a is a fixed integer. If the same question is considered for congruences xn ≡ 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 diphantine geometry.
Kenneth Ireland, Michael Rosen
Chapter 12. Algebraic Number Theory
Abstract
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 Fermat’s 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(ζp). 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 will define 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 Chapter 10 we discussed Diophantine equations over finite fields. 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 come 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
Backmatter
Metadaten
Titel
A Classical Introduction to Modern Number Theory
verfasst von
Kenneth Ireland
Michael Rosen
Copyright-Jahr
1982
Verlag
Springer New York
Electronic ISBN
978-1-4757-1779-2
Print ISBN
978-1-4757-1781-5
DOI
https://doi.org/10.1007/978-1-4757-1779-2