NUMBER THEORY IN ALGEBRA
CONTENTS PART I - ELEMENTS OF NUMBER THEORY AND ALGEBRA 1 THEOREMS OF EULER, FERMAT AND LAGRANGE HISTORICAL PERSPECTIVE 1 1 INTRODUCTION 1 2 THE QUOTIENT RING Z/RZ 1 3 AN ELEMENTARY COUNTING PRINCIPLE EXERCISES REFERENCES 5 THE CHINESE REMAINDER THEOREM AND THE EVALUATION OF NUMBER OF SOLUTIONS OF A LINEAR CONGRUENCE WITH SIDE CONDITIONS 5 1 INTRODUCTION 5 2 THE CHINESE REMAINDER THEOREM 5 3 DIRECT PRODUCTS AND DIRECT SUMS 5 4 EVEN FUNCTIONS (MOD R) 5 5 LINEAR CONGRUENCES WITH SIDE CONDITIONS 5 6 THE RADEMACHER FORMULA 5 7 NOTES WITH ILLUSTRATIVE EXAMPLES 5 8 WORKED-OUT EXAMPLES 6 RECIPROCITY LAWS 6 1 INTRODUCTION 6 2 PRELIMINARIES 6 3 GAUSS LEMMA 6 4 FINITE ?ELDS AND QUADRATIC RECIPROCITY LAW 6 5 CUBIC RESIDUES (MOD P) 6 6 GROUP CHARACTERS AND THE CUBIC RECIPROCITY LAW 6 7 NOTES WITH ILLUSTRATIVE EXAMPLES 6 8 A COMMENT BY W C WATERHOUSE 6 9 WORKED-OUT EXAMPLES 7 FINITE GROUPS 7 1 INTRODUCTION 7 2 CONJUGATE CLASSES OF ELEMENTS IN A GROUP 7 3 COUNTING CERTAIN SPECIAL REPRESENTATIONS OF A GROUP ELEMENT 7 4 NUMBER OF CYCLIC SUBGROUPS OF A ?NITE GROUP 7 5 A CRITERION FOR THE UNIQUENESS OF A CYCLIC GROUP OF ORDER R 7 6 NOTES WITH ILLUSTRATIVE EXAMPLES 7 7 A WORKED-OUT EXAMPLE 7 8 AN EXAMPLE FROM QUADRATIC RESIDUES PART II - THE RELEVANCE OF ALGEBRAIC STRUCTURES TO NUMBER THEORY 8 ORDERED ?ELDS, ?ELDS WITH VALUATION AND OTHER ALGEBRAIC STRUCTURES 8 1 INTRODUCTION 8 2 ORDERED ?ELDS 8 3 VALUATION RINGS 8 4 FIELDS WITH VALUATION 8 5 NORMED DIVISION DOMAINS 8 6 MODULAR LATTICES AND JORDAN-HÖLDER THEOREM 8 7 NON-COMMUTATIVE RINGS 8 8 BOOLEAN ALGEBRAS 8 9 NOTES WITH ILLUSTRATIVE EXAMPLES 8 10 WORKED-OUT EXAMPLES 9 THE ROLE OF THE MÖBIUS FUNCTION— ABSTRACT MÖBIUS INVERSION 9 1 INTRODUCTION 9 2 ABSTRACT MÖBIUS INVERSION 9 3 INCIDENCE ALGEBRA OF N×N MATRICES 9 4 VECTOR SPACES OVER A ?NITE ?ELD 9 5 NOTES WITH ILLUSTRATIVE EXAMPLES 9 6 WORKED-OUT EXAMPLES 10 THE ROLE OF GENERATING FUNCTIONS 10 1 INTRODUCTION 10 2 EULER’S THEOREMS ON PARTITIONS OF AN INTEGER 10 3 ELLIPTIC FUNCTIONS 10 4 STIRLING NUMBERS AND BERNOULLI NUMBERS 10 5 BINOMIAL POSETS AND GENERATING FUNCTIONS 10 6 DIRICHLET SERIES 10 7 NOTES WITH ILLUSTRATIVE EXAMPLES 10 8 WORKED-OUT EXAMPLES 10 9 CATALAN NUMBERS 11 SEMIGROUPS AND CERTAIN CONVOLUTION ALGEBRAS 11 1 INTRODUCTION 11 2 SEMIGROUPS 11 3 SEMICHARACTERS 11 4 FINITE DIMENSIONAL CONVOLUTION ALGEBRAS 11 5 ABSTRACT ARITHMETICAL FUNCTIONS 11 6 CONVOLUTIONS IN GENERAL 11 7 A FUNCTIONAL-THEORETIC ALGEBRA 11 8 NOTES WITH ILLUSTRATIVE EXAMPLES 11 9 WORKED-OUT EXAMPLES PART III - A GLIMPSE OF ALGEBRAIC NUMBER THEORY 12 NOETHERIAN AND DEDEKIND DOMAINS 12 1 INTRODUCTION 12 2 NOETHERIAN RINGS 12 3 MORE ABOUT IDEALS 12 4 JACOBSON RADICAL 12 5 THE LASKER–NOETHER DECOMPOSITION THEOREM 12 6 DEDEKIND DOMAINS 12 7 THE CHINESE REMAINDER THEOREM REVISITED 12 8 INTEGRAL DOMAINS HAVING ?NITE NORM PROPERTY 12 9 NOTES WITH ILLUSTRATIVE EXAMPLES 12 10 WORKED-OUT EXAMPLES 13 ALGEBRAIC NUMBER ?ELDS 13 1 INTRODUCTION 13 2 THE IDEAL CLASS GROUP 13 3 CYCLOTOMIC ?ELDS 13 4 HALF-FACTORIAL DOMAINS 13 5 THE PELL EQUATION 13 6 THE CAKRAVALA METHOD 13 7 DIRICHLET’S UNIT THEOREM 13 8 NOTES WITH ILLUSTRATIVE EXAMPLES 13 9 FORMALLY REAL ?ELDS 13 10 WORKED-OUT EXAMPLES PART IV - SOME MORE INTERCONNECTIONS 14 RINGS OF ARITHMETIC FUNCTIONS 14 1 INTRODUCTION 14 2 CAUCHY COMPOSITION (MOD R) 14 3 THE ALGEBRA OF EVEN FUNCTIONS (MOD R) 14 4 CARLITZ CONJECTURE 14 5 MORE ABOUT ZERO DIVISORS 14 6 CERTAIN NORM-PRESERVING TRANSFORMATIONS 14 7 NOTES WITH ILLUSTRATIVE EXAMPLES 14 8 WORKED-OUT EXAMPLES 15 ANALOGUES OF THE GOLDBACH PROBLEM 15 1 INTRODUCTION 15 2 THE RIEMANN HYPOTHESIS 15 3 A ?NITE ANALOGUE OF THE GOLDBACH PROBLEM 15 4 THE GOLDBACH PROBLEM IN MN(Z) 15 5 AN ANALOGUE OF GOLDBACH THEOREM VIA POLYNOMIALS OVER ?NITE ?ELDS 15 6 NOTES WITH ILLUSTRATIVE EXAMPLES 15 7 A VARIANT OF GOLDBACH CONJECTURE 16 AN EPILOGUE: MORE INTERCONNECTIONS INTRODUCTION 16 1 ON COMMUTATIVE RINGS 16 2 COMMUTATIVE RINGS WITHOUT MAXIMAL IDEALS 16 3 IN?NITUDE OF PRIMES IN A PID 16 4 ON THE GROUP OF UNITS OF A COMMUTATIVE RING 16 5 QUADRATIC RECIPROCITY IN A ?NITE GROUP 16 6 WORKED-OUT EXAMPLES TRUE/FALSE STATEMENTS : ANSWER KEY