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## Über dieses Buch

This book corresponds to a mathematical course given in 1986/87 at the University Louis Pasteur, Strasbourg. This work is primarily intended for graduate students. The following are necessary prerequisites : a few standard definitions in set theory, the definition of rational integers, some elementary facts in Combinatorics (maybe only Newton's binomial formula), some theorems of Analysis at the level of high schools, and some elementary Algebra (basic results about groups, rings, fields and linear algebra). An important place is given to exercises. These exercises are only rarely direct applications of the course. More often, they constitute complements to the text. Mostly, hints or references are given so that the reader should be able to find solutions. Chapters one and two deal with elementary results of Number Theory, for example : the euclidean algorithm, the Chinese remainder theorem and Fermat's little theorem. These results are useful by themselves, but they also constitute a concrete introduction to some notions in abstract algebra (for example, euclidean rings, principal rings ... ). Algorithms are given for arithmetical operations with long integers. The rest of the book, chapters 3 through 7, deals with polynomials. We give general results on polynomials over arbitrary rings. Then polynomials with complex coefficients are studied in chapter 4, including many estimates on the complex roots of polynomials. Some of these estimates are very useful in the subsequent chapters.

## Inhaltsverzeichnis

### Chapter 1. Elementary Arithmetics

Abstract
This chapter deals with the representation of rational integers and describes the basic algorithms of arithmetics (addition, multiplication, division,...). The study of theses examples introduces the fundamental notion of the cost of an algorithm
Maurice Mignotte

### Chapter 2. Number Theory, Complements

Abstract
This chapter contains some results of elementary number theory, which will be useful later and in some applications not considered here, such as cryptography.
Maurice Mignotte

### Chapter 3. Polynomials, Algebraic Study

Abstract
This chapter presents definitions and general algebraic properties of polynomials over arbitrary domains.
Maurice Mignotte

### Chapter 4. Polynomials with Complex Coefficients

Abstract
The main topic of this chapter is the study of the zeros of polynomials with complex coefficients. This study leads to inequalities about the size of factors of polynomials. These inequalities play an important rôle in the last two chapters.
Maurice Mignotte

### Chapter 5. Polynomials with Real Coefficients

Abstract
In this chapter, we call an ordered field such that the field $$\mathbb{C} = \mathbb{R}[\sqrt {{ - 1}} ]$$ is algebraically closed. Usually, we consider that E is the field of real numbers and that C is the field of complex numbers. We study algorithms to separate the real roots of polynomials.
Maurice Mignotte

### Chapter 6. Polynomials Over Finite Fields

Abstract
The first step of modern algorithms of factorization of polynomials with integer coefficients consists in factorizing their image modulo some prime number. This is the reason why, in this chapter, we study the factorization of polynomials over finite fields. Most of the results of this theory were developed by E.R. Berlekamp.
Maurice Mignotte

### Chapter 7. Polynomials with Integer Coefficients

Abstract
This chapter is the culmination of this book. Here we describe algorithms of factorization of polynomials with integer coefficients. These algorithms use many of the results demonstrated in the previous chapters.
Maurice Mignotte

### Backmatter

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