FUNCTIONS AND RELATIONS

By Q. KAVIR

CONTENTS SET THEORY SETS AND FUNCTIONS THE CARTESIAN PRODUCT; OPERATIONS; RELATIONS EQUIVALENCE RELATIONS PARTIAL ORDERS PROJECTS NUMBER THEORY BASIC PROPERTIES OF INTEGERS MODULAR ARITHMETIC MATHEMATICAL INDUCTION GROUPS SYMMETRIES OF THE REGULAR N-GON INTRODUCTION TO GROUPS PROPERTIES OF GROUP ELEMENTS SYMMETRIC GROUPS SUBGROUPS LATTICE OF SUBGROUPS GROUP HOMOMORPHISMS GROUP PRESENTATIONS GROUPS IN GEOMETRY DI?E-HELLMAN PUBLIC KEY SEMIGROUPS AND MONOIDS QUOTIENT GROUPS COSETS AND LAGRANGE’S THEOREM CONJUGACY AND NORMAL SUBGROUPS ISOMORPHISM THEOREMS FUNDAMENTAL THEOREM OF FINITELY GENERATED ABELIAN GROUPS RINGS INTRODUCTION TO RINGS RINGS GENERATED BY ELEMENTS MATRIX RINGS RING HOMOMORPHISMS IDEALS QUOTIENT RINGS MAXIMAL IDEALS AND PRIME IDEALS DIVISIBILITY IN COMMUTATIVE RINGS RINGS OF FRACTIONS EUCLIDEAN DOMAINS UNIQUE FACTORIZATION DOMAINS FACTORIZATION OF POLYNOMIALS RSA CRYPTOGRAPHY ALGEBRAIC INTEGERS FIELD EXTENSIONS INTRODUCTION TO FIELD EXTENSIONS ALGEBRAIC EXTENSIONS SOLVING CUBIC AND QUARTIC EQUATIONS CONSTRUCTIBLE NUMBERS CYCLOTOMIC EXTENSIONS SPLITTING FIELDS AND ALGEBRAIC CLOSURE FINITE FIELDS GROUP ACTIONS INTRODUCTION TO GROUP ACTIONS ORBITS AND STABILIZERS TRANSITIVE GROUP ACTIONS GROUPS ACTING ON THEMSELVES SYLOW’S THEOREM A BRIEF INTRODUCTION TO REPRESENTATIONS OF GROUPS CLASSI?CATION OF GROUPS COMPOSITION SERIES AND SOLVABLE GROUPS FINITE SIMPLE GROUPS SEMIDIRECT PRODUCT CLASSI?CATION THEOREMS NILPOTENT GROUPS 10 MODULES AND ALGEBRAS BOOLEAN ALGEBRAS VECTOR SPACES INTRODUCTION TO MODULES HOMOMORPHISMS AND QUOTIENT MODULES FREE MODULES AND MODULE DECOMPOSITION FINITELY GENERATED MODULES OVER PIDS, I FINITELY GENERATED MODULES OVER PIDS, II APPLICATIONS TO LINEAR TRANSFORMATIONS JORDAN CANONICAL FORM 10 10 APPLICATIONS OF THE JORDAN CANONICAL FORM 10 11 A BRIEF INTRODUCTION TO PATH ALGEBRAS 10 12 PROJECTS 11 GALOIS THEORY AUTOMORPHISMS OF FIELD EXTENSIONS FUNDAMENTAL THEOREM OF GALOIS THEORY FIRST APPLICATIONS OF GALOIS THEORY GALOIS GROUPS OF CYCLOTOMIC EXTENSIONS SYMMETRIES AMONG ROOTS; THE DISCRIMINANT COMPUTING GALOIS GROUPS OF POLYNOMIALS FIELDS OF FINITE CHARACTERISTIC SOLVABILITY BY RADICALS 12 MULTIVARIABLE POLYNOMIAL RINGS INTRODUCTION TO NOETHERIAN RINGS MULTIVARIABLE POLYNOMIALS AND A?NE SPACE THE NULLSTELLENSATZ POLYNOMIAL DIVISION; MONOMIAL ORDERS GR¨OBNER BASES BUCHBERGER’S ALGORITHM APPLICATIONS OF GR¨OBNER BASES A BRIEF INTRODUCTION TO ALGEBRAIC GEOMETRY 13 CATEGORIES INTRODUCTION TO CATEGORIES FUNCTORS