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Full Description
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.
Contents
1 Elementary Arithmetics.- 1. Representation of an integer in basis B1.- 2. Addition.- 3. Subtraction.- 4. Multiplication.- 5. Euclidean division.- 6. The cost of multiplication and division.- 7. How to compute powers.- 8. The g.c.d..- 9. The group G (n).- 10. The Chinese remainder theorem.- 11. The prime numbers.- 2 Number Theory, Complements.- 1. Study of the group G(n).- 2. Tests of primality.- 3. Factorization of rational integers.- 3 Polynomials, Algebraic Study.- 1. Definitions and elementary properties.- 2. Euclidean division.- 3. The Chinese remainder theorem.- 4. Factorization.- 5. Polynomial functions.- 6. The resultant.- 7. Companion matrix.- 8. Linear recursive sequences.- 4 Polynomials with complex coefficients.- 1. The theorem of d'Alembert.- 2. Estimates of the roots.- 3. The measure of a polynomial.- 4. Bounds for size of the factors of a polynomial.- 5. The distribution of the roots of a polynomial.- 6. Separation of the roots of a polynomial.- 5 Polynomials with real coefficients.- 1. Polynomials irreducible over ?.- 2. The theorem of Rolle.- 3. Estimates of real roots.- 4. The number of zeros of a polynomial in a real interval.- 5. Equations whose roots have a negative real part.- 6/Polynomials over finite fields.- 1. Finite fields.- 2. Statistics on Hq[X].- 3. Factorization into a product of squarefree polynomials.- 4. Factorization of the polynomials over a finite field.- 5. Search for the roots of a polynomial in a finite field.- 7 Polynomials with integer coefficients.- 1. Principles of the algorithms of factorization.- 2. The choice of the prime modulus.- 3. Refining the factorization.- 4. Berlekamp's method of factorization.- 5. The algorithm L3.- 6. Factors of polynomials and lattices.- 7. The algorithm of factorization.- Index of Names.
