HIGHER ALGEBRA

By Y. CAINI

CONTENTS PART I NUMBERS NUMBERS INDUCTION EUCLID’S ALGORITHM UNIQUE FACTORIZATION CONGRUENCE PART II CONGRUENCE CLASSES AND RINGS CONGRUENCE CLASSES RINGS AND FIELDS MATRICES AND CODES PART III CONGRUENCES AND GROUPS FERMAT’S AND EULER’S THEOREMS APPLICATIONS OF EULER’S THEOREM GROUPS THE CHINESE REMAINDER THEOREM PART IV POLYNOMIALS POLYNOMIALS CONTENTS THE FUNDAMENTAL THEOREM OF ALGEBRA POLYNOMIALS IN Q[X] CONGRUENCES AND THE CHINESE REMAINDER THEOREM FAST POLYNOMIAL MULTIPLICATION PART V PRIMITIVE ROOTS CYCLIC GROUPS AND CRYPTOGRAPHY CARMICHAEL NUMBERS QUADRATIC RECIPROCITY QUADRATIC APPLICATIONS PART VI FINITE FIELDS CONGRUENCE CLASSES MODULO A POLYNOMIAL HOMOMORPHISMS AND FINITE FIELDS BCH CODES PART VII FACTORING POLYNOMIALS FACTORING IN Z[X] IRREDUCIBLE POLYNOMIALS ANSWERS AND HINTS TO THE EXERCISES REFERENCES