MODERN ALGEBRA WITH APPLICATIONS
CONTENTS INTRODUCTION CLASSICAL ALGEBRA, MODERN ALGEBRA, BINARY OPERATIONS, ALGEBRAIC STRUCTURES, EXTENDING NUMBER SYSTEMS, BOOLEAN ALGEBRAS ALGEBRA OF SETS, NUMBER OF ELEMENTS IN A SET, BOOLEAN ALGEBRAS, PROPOSITIONAL LOGIC, SWITCHING CIRCUITS, DIVISORS, POSETS AND LATTICES, NORMAL FORMS AND SIMPLI?CATION OF CIRCUITS, TRANSISTOR GATES, REPRESENTATION THEOREM, EXERCISES, GROUPS GROUPS AND SYMMETRIES, SUBGROUPS, CYCLIC GROUPS AND DIHEDRAL GROUPS, MORPHISMS, PERMUTATION GROUPS, EVEN AND ODD PERMUTATIONS, CAYLEY’S REPRESENTATION THEOREM, QUOTIENT GROUPS EQUIVALENCE RELATIONS, COSETS AND LAGRANGE’S THEOREM, NORMAL SUBGROUPS AND QUOTIENT GROUPS, MORPHISM THEOREM, DIRECT PRODUCTS, GROUPS OF LOW ORDER, ACTION OF A GROUP ON A SET, SYMMETRY GROUPS IN THREE DIMENSIONS TRANSLATIONS AND THE EUCLIDEAN GROUP, MATRIX GROUPS, FINITE GROUPS IN TWO DIMENSIONS, PROPER ROTATIONS OF REGULAR SOLIDS, FINITE ROTATION GROUPS IN THREE DIMENSIONS, CRYSTALLOGRAPHIC GROUPS, P´OLYA–BURNSIDE METHOD OF ENUMERATION BURNSIDE’S THEOREM, NECKLACE PROBLEMS, COLORING POLYHEDRA, COUNTING SWITCHING CIRCUITS, MONOIDS AND MACHINES MONOIDS AND SEMIGROUPS, FINITE-STATE MACHINES, QUOTIENT MONOIDS AND THE MONOID OF A MACHINE, RINGS AND FIELDS RINGS, INTEGRAL DOMAINS AND FIELDS, SUBRINGS AND MORPHISMS OF RINGS, NEW RINGS FROM OLD, FIELD OF FRACTIONS, CONVOLUTION FRACTIONS, POLYNOMIAL AND EUCLIDEAN RINGS EUCLIDEAN RINGS, EUCLIDEAN ALGORITHM, UNIQUE FACTORIZATION, FACTORING REAL AND COMPLEX POLYNOMIALS, FACTORING RATIONAL AND INTEGRAL POLYNOMIALS, FACTORING POLYNOMIALS OVER FINITE FIELDS, LINEAR CONGRUENCES AND THE CHINESE REMAINDER THEOREM, QUOTIENT RINGS IDEALS AND QUOTIENT RINGS, COMPUTATIONS IN QUOTIENT RINGS, QUOTIENT POLYNOMIAL RINGS THAT ARE FIELDS, FIELD EXTENSIONS FIELD EXTENSIONS, ALGEBRAIC NUMBERS, GALOIS FIELDS, PRIMITIVE ELEMENTS, LATIN SQUARES LATIN SQUARES, ORTHOGONAL LATIN SQUARES, FINITE GEOMETRIES, MAGIC SQUARES, GEOMETRICAL CONSTRUCTIONS CONSTRUCTIBLE NUMBERS, DUPLICATING A CUBE, TRISECTING AN ANGLE, SQUARING THE CIRCLE, CONSTRUCTING REGULAR POLYGONS, NONCONSTRUCTIBLE NUMBER OF DEGREE 4, ERROR-CORRECTING CODES THE CODING PROBLEM, SIMPLE CODES, POLYNOMIAL REPRESENTATION, MATRIX REPRESENTATION, ERROR CORRECTING AND DECODING, BCH CODES,