ALGEBRA: ABSTRACT AND CONCRETE
CONTENTS CHAPTER 1 ALGEBRAIC THEMES 1 1 WHAT IS SYMMETRY? 1 2 SYMMETRIES OF THE RECTANGLE AND THE SQUARE 1 3 MULTIPLICATION TABLES 1 4 SYMMETRIES AND MATRICES 1 5 PERMUTATIONS 1 6 DIVISIBILITY IN THE INTEGERS 1 7 MODULAR ARITHMETIC 1 8 POLYNOMIALS 1 9 COUNTING 1 10 GROUPS 1 11 RINGS AND FIELDS 1 12 AN APPLICATION TO CRYPTOGRAPHY CHAPTER 2 BASIC THEORY OF GROUPS 2 1 FIRST RESULTS 2 2 SUBGROUPS AND CYCLIC GROUPS 2 3 THE DIHEDRAL GROUPS 2 4 HOMOMORPHISMS AND ISOMORPHISMS 2 5 COSETS AND LAGRANGE’S THEOREM 2 6 EQUIVALENCE RELATIONS AND SET PARTITIONS 2 7 QUOTIENT GROUPS AND HOMOMORPHISM THEOREMS CHAPTER 3 PRODUCTS OF GROUPS 3 1 DIRECT PRODUCTS 3 2 SEMIDIRECT PRODUCTS 3 3 VECTOR SPACES 3 4 THE DUAL OF A VECTOR SPACE AND MATRICES 3 5 LINEAR ALGEBRA OVER Z 3 6 FINITELY GENERATED ABELIAN GROUPS CHAPTER 4 SYMMETRIES OF POLYHEDRA CONTENTS 4 1 ROTATIONS OF REGULAR POLYHEDRA 4 2 ROTATIONS OF THE DODECAHEDRON AND ICOSAHEDRON 4 3 WHAT ABOUT RE?ECTIONS? 4 4 LINEAR ISOMETRIES 4 5 THE FULL SYMMETRY GROUP AND CHIRALITY CHAPTER 5 ACTIONS OF GROUPS 5 1 GROUP ACTIONS ON SETS 5 2 GROUP ACTIONS—COUNTING ORBITS 5 3 SYMMETRIES OF GROUPS 5 4 GROUP ACTIONS AND GROUP STRUCTURE 5 6 ADDITIONAL EXERCISES FOR CHAPTER CHAPTER 6 RINGS 6 1 A RECOLLECTION OF RINGS 6 2 HOMOMORPHISMS AND IDEALS 6 3 QUOTIENT RINGS 6 4 INTEGRAL DOMAINS 6 5 EUCLIDEAN DOMAINS, PRINCIPAL IDEAL DOMAINS, AND UNIQUE FACTORIZATION 6 6 UNIQUE FACTORIZATION DOMAINS 6 7 NOETHERIAN RINGS 6 8 IRREDUCIBILITY CRITERIA CHAPTER 7 FIELD EXTENSIONS – FIRST LOOK 7 1 A BRIEF HISTORY 7 2 SOLVING THE CUBIC EQUATION 7 3 ADJOINING ALGEBRAIC ELEMENTS TO A FIELD 7 4 SPLITTING FIELD OF A CUBIC POLYNOMIAL 7 5 SPLITTING FIELDS OF POLYNOMIALS IN CŒX CHAPTER 8 MODULES 8 1 THE IDEA OF A MODULE 8 2 HOMOMORPHISMS AND QUOTIENT MODULES 8 3 MULTILINEAR MAPS AND DETERMINANTS 8 4 FINITELY GENERATED MODULES OVER A PID, PART I 8 5 FINITELY GENERATED MODULES OVER A PID, PART II 8 6 RATIONAL CANONICAL FORM 8 7 JORDAN CANONICAL FORM CHAPTER 9 FIELD EXTENSIONS – SECOND LOOK 9 1 FINITE AND ALGEBRAIC EXTENSIONS 9 2 SPLITTING FIELDS 9 3 THE DERIVATIVE AND MULTIPLE ROOTS CONTENTS 9 4 SPLITTING FIELDS AND AUTOMORPHISMS 9 5 THE GALOIS CORRESPONDENCE 9 6 SYMMETRIC FUNCTIONS 9 7 THE GENERAL EQUATION OF DEGREE N 9 8 QUARTIC POLYNOMIALS 9 9 GALOIS GROUPS OF HIGHER DEGREE POLYNOMIALS CHAPTER 10 SOLVABILITY 10 1 COMPOSITION SERIES AND SOLVABLE GROUPS 10 2 COMMUTATORS AND SOLVABILITY 10 3 SIMPLICITY OF THE ALTERNATING GROUPS 10 4 CYCLOTOMIC POLYNOMIALS 10 5 THE EQUATION XN B D 10 6 SOLVABILITY BY RADICALS 10 7 RADICAL EXTENSIONS CHAPTER 11 ISOMETRY GROUPS 11 1 MORE ON ISOMETRIES OF EUCLIDEAN SPACE 11 2 EULER’S THEOREM 11 3 FINITE ROTATION GROUPS 11 4 CRYSTALS