Number Theory and Its Applications

Fuhuo Li, Nianliang Wang, Shigeru Kanemitsu

• Publisher: WSPC
• Year: 2012
• ISBN:
  9789814425650
• Topic: Mathématiques
• Subtopic: Théorie des nombres

This book emphasizes the role of symmetry and presents as many viewpoints as possible of an important phenomenon — the functional equation of the associated zeta-function. It starts from the basics before warping into the space of new interest; from the ground state to the excited state. For example, the Euler function is treated in several different places, as the number of generators of a finite cyclic group, as one counting the order of the multiplicative group of reduced residue classes modulo q, and as the order and degree of the Galois group of the cyclotomic field, respectively.

One of the important principles of learning is to work with the material many times. This book presents many worked-out examples and exercises to enhance the reader's comprehension on the topics covered in an in-depth manner. This is done in a different setting each time such that the reader will always be challenged. For the keen reader, even browsing the text alone, without solving the exercises, will yield some knowledge and enjoyment.

Contents:

• Elements of Algebra
• Rudiments of Algebraic Number Theory
• Arithmetical Functions and Stieltjes Integrals
• Quadratic Reciprocity Through Duality
• Around Dirichlet L -Functions
• Control Systems and Number Theory

Readership: Advanced undergraduate students, graduate students and researchers interested in algebra through number-theoretic examples, number-theoretic control systems, etc.

Chapter 1

Elements of algebra

In this chapter we make a brief description of modern algebra to such an extent that is necessary to make algebraic elucidation of some important materials in elementary number theory which are usually treated in an intuitive and elementary way. Algebraic preliminaries can be found in many textbooks (cf. e.g. [Hatt]) but our presentation gives a quick introduction through groups. We recall elements of algebra, groups, rings (Euclidean domains) and fields including the notion of orbits. These are then applied to deducing that the ring of rational integers is a UFD (Euclidean → PID → UFD), a generalization of Fermat's little theorem, which then gives rise to Fermat's little theorem, etc. These abstract notions are also used in Chapter 2 in the case of an algebraic number field, thus generalizing elementary number theory to algebraic number theory. This chapter can serve as a mini-course on abstract algebra for undergraduates.

1.1 Preliminaries

The notion of equivalence classes is one of the most fundamental in modern mathematics and in daily life matters as well.

Definition 1.1. A (binary) relation ~ defined on a set X ≠ Ø is called an equivalence relation if it satisfies

(i) (reflexive law) x ~ x.

(ii) (symmetric law) If x ~ y, then y ~ x.

(iii) (transitive law) If x ~ y and y ~ z, then x ~ z.

If x ~ y, then x is said to be equivalent to y or conversely in view of (ii). The set C_x of all elements y equivalent to x is called an equivalence class containing x:

The following theorem provides us with a classification X / ~ of the set X into disjoint union of mutually inequivalent classes.

Theorem 1.1. The following statements ar...