BASIC ALGEBRA

By F. PAANE

CONTENTS I PRELIMINARIES ABOUT THE INTEGERS, POLYNOMIALS, AND MATRICES 1 DIVISION AND EUCLIDEAN ALGORITHMS 2 UNIQUE FACTORIZATION OF INTEGERS 3 UNIQUE FACTORIZATION OF POLYNOMIALS 4 PERMUTATIONS AND THEIR SIGNS 5 ROW REDUCTION 6 MATRIX OPERATIONS 7 PROBLEMS II VECTOR SPACES OVER Q, R, AND C 1 SPANNING, LINEAR INDEPENDENCE, AND BASES 2 VECTOR SPACES DE?NED BY MATRICES 3 LINEAR MAPS 4 DUAL SPACES 5 QUOTIENTS OF VECTOR SPACES 6 DIRECT SUMS AND DIRECT PRODUCTS OF VECTOR SPACES 7 DETERMINANTS 8 EIGENVECTORS AND CHARACTERISTIC POLYNOMIALS 9 BASES IN THE IN?NITE-DIMENSIONAL CASE 10 PROBLEMS III INNER-PRODUCT SPACES 1 INNER PRODUCTS AND ORTHONORMAL SETS 2 ADJOINTS 3 SPECTRAL THEOREM 4 PROBLEMS IV GROUPS AND GROUP ACTIONS 1 GROUPS AND SUBGROUPS 2 QUOTIENT SPACES AND HOMOMORPHISMS 3 DIRECT PRODUCTS AND DIRECT SUMS 4 RINGS AND FIELDS 5 POLYNOMIALS AND VECTOR SPACES 6 GROUP ACTIONS AND EXAMPLES 7 SEMIDIRECT PRODUCTS 8 SIMPLE GROUPS AND COMPOSITION SERIES 9 STRUCTURE OF FINITELY GENERATED ABELIAN GROUPS 10 SYLOW THEOREMS 11 CATEGORIES AND FUNCTORS 12 PROBLEMS V THEORY OF A SINGLE LINEAR TRANSFORMATION 1 INTRODUCTION 2 DETERMINANTS OVER COMMUTATIVE RINGS WITH IDENTITY 3 CHARACTERISTIC AND MINIMAL POLYNOMIALS 4 PROJECTION OPERATORS 5 PRIMARY DECOMPOSITION 6 JORDAN CANONICAL FORM 7 COMPUTATIONS WITH JORDAN FORM 8 PROBLEMS VI MULTILINEAR ALGEBRA 1 BILINEAR FORMS AND MATRICES 2 SYMMETRIC BILINEAR FORMS 3 ALTERNATING BILINEAR FORMS 4 HERMITIAN FORMS 5 GROUPS LEAVING A BILINEAR FORM INVARIANT 6 TENSOR PRODUCT OF TWO VECTOR SPACES 7 TENSOR ALGEBRA 8 SYMMETRIC ALGEBRA 9 EXTERIOR ALGEBRA VII ADVANCED GROUP THEORY 1 FREE GROUPS 2 SUBGROUPS OF FREE GROUPS 3 FREE PRODUCTS 4 GROUP REPRESENTATIONS VII ADVANCED GROUP THEORY (CONTINUED) 5 BURNSIDE’S THEOREM 6 EXTENSIONS OF GROUPS VIII COMMUTATIVE RINGS AND THEIR MODULES 1 EXAMPLES OF RINGS AND MODULES 2 INTEGRAL DOMAINS AND FIELDS OF FRACTIONS 3 PRIME AND MAXIMAL IDEALS 4 UNIQUE FACTORIZATION 5 GAUSS’S LEMMA 6 FINITELY GENERATED MODULES 7 ORIENTATION FOR ALGEBRAIC NUMBER THEORY AND ALGEBRAIC GEOMETRY 8 NOETHERIAN RINGS AND THE HILBERT BASIS THEOREM 9 INTEGRAL CLOSURE 10 LOCALIZATION AND LOCAL RINGS 11 DEDEKIND DOMAINS IX FIELDS AND GALOIS THEORY 1 ALGEBRAIC ELEMENTS 2 CONSTRUCTION OF FIELD EXTENSIONS 3 FINITE FIELDS 4 ALGEBRAIC CLOSURE 5 GEOMETRIC CONSTRUCTIONS BY STRAIGHTEDGE AND COMPASS 6 SEPARABLE EXTENSIONS 7 NORMAL EXTENSIONS 8 FUNDAMENTAL THEOREM OF GALOIS THEORY 9 APPLICATION TO CONSTRUCTIBILITY OF REGULAR POLYGONS 10 APPLICATION TO PROVING THE FUNDAMENTAL THEOREM OF ALGEBRA 11 APPLICATION TO UNSOLVABILITY OF POLYNOMIAL EQUATIONS WITH NONSOLVABLE GALOIS GROUP 12 CONSTRUCTION OF REGULAR POLYGONS 13 SOLUTION OF CERTAIN POLYNOMIAL EQUATIONS WITH SOLVABLE GALOIS GROUP 14 PROOF THAT ? IS TRANSCENDENTAL 15 NORM AND TRACE 16 SPLITTING OF PRIME IDEALS IN EXTENSIONS 17 TWO TOOLS FOR COMPUTING GALOIS GROUPS 18 PROBLEMS X MODULES OVER NONCOMMUTATIVE RINGS 1 SIMPLE AND SEMISIMPLE MODULES 2 COMPOSITION SERIES 3 CHAIN CONDITIONS 4 HOM AND END FOR MODULES 5 TENSOR PRODUCT FOR MODULES 6 EXACT SEQUENCES