ALGEBRA OF POLYNOMIALS
CONTENTS CHAPTER 1 THE INTEGERS 1 1 BASIC PROPERTIES 1 2 WELL ORDERING PRINCIPLE 1 3 PRIMES 1 4 MANY PRIMES 1 5 EUCLIDEAN ALGORITHM 1 6 FACTORING INTEGERS 1 7 IRRATIONAL NUMBERS 1 8 UNIQUE FACTORIZATION IN MORE GENERAL RINGS NOTES ON CHAPTER CHAPTER 2 MODULAR ARITHMETIC 2 1 LINEAR EQUATIONS 2 2 CONGRUENCES 2 3 THE RING ZN 2 4 EQUIVALENCE RELATIONS 2 5 CHINESE REMAINDER THEOREM 2 6 CONGRUENCE EQUATIONS 2 7 FERMAT’S LITTLE THEOREM 2 8 EULER’S THEOREM 2 9 MORE ON EULER’S PHI FUNCTION 2 10 PRIMITIVE ROOTS CHAPTER 3 DIOPHANTINE EQUATIONS AND QUADRATIC NUMBER DOMAINS 3 1 PYTHAGOREAN TRIPLES 3 2 FERMAT’S EQUATION FOR N = 3 3 QUADRATIC NUMBER DOMAINS 3 4 PELL’S EQUATION 3 5 THE GAUSSIAN INTEGERS 3 6 QUADRATIC RECIPROCITY CHAPTER 4 CODES AND FACTORING 4 1 CODES 4 2 THE RIVEST-SHAMIR-ADELMAN SCHEME 4 3 PRIMALITY TESTING 4 4 FACTORING ALGORITHMS CHAPTER 5 REAL AND COMPLEX NUMBERS 5 1 REAL NUMBERS 5 2 COMPLEX NUMBERS 5 3 POLAR FORM 5 4 THE EXPONENTIAL FUNCTION 5 5 FUNDAMENTAL THEOREM OF ALGEBRA 5 6 REAL POLYNOMIALS CHAPTER 6 THE RING OF POLYNOMIALS 6 1 PRELIMINARIES ON POLYNOMIALS 6 2 UNIQUE FACTORIZATION FOR POLYNOMIALS 6 3 IRREDUCIBLE POLYNOMIALS IN Z[X] 6 4 EISENSTEIN’S CRITERION 6 5 FACTORING MODULO PRIMES 6 6 ALGEBRAIC NUMBERS 6 7 TRANSCENDENTAL NUMBERS 6 8 STURM’S ALGORITHM 6 9 SYMMETRIC FUNCTIONS 6 10 CUBIC POLYNOMIALS CHAPTER 7 FINITE FIELDS 7 1 ARITHMETIC MODULO A POLYNOMIAL 7 2 AN EIGHT-ELEMENT FIELD 7 3 FERMAT’S LITTLE THEOREM FOR FINITE FIELDS 7 4 CHARACTERISTIC 7 5 ALGEBRAIC ELEMENTS 7 6 FINITE FIELDS 7 7 AUTOMORPHISMS OF FPD 7 8 IRREDUCIBLE POLYNOMIALS OF ALL DEGREES 7 9 FACTORING ALGORITHMS FOR POLYNOMIALS 7 10 FACTORING RATIONAL POLYNOMIALS