ABSTRACT ALGEBRA

By D. KUENK

PREREQUISITES 0 1 ELEMENTARY NUMBER THEORY 0 2 SET THEORY 0 3 LINEAR ALGEBRA CHAPTER GROUP FUNDAMENTALS 1 1 GROUPS AND SUBGROUPS 1 2 PERMUTATION GROUPS 1 3 COSETS, NORMAL SUBGROUPS AND HOMOMORPHISMS 1 4 THE ISOMORPHISM THEOREMS 1 5 DIRECT PRODUCTS CHAPTER RING FUNDAMENTALS 2 1 BASIC DE?NITIONS AND PROPERTIES 2 2 IDEALS, HOMOMORPHISMS AND QUOTIENT RINGS 2 3 THE ISOMORPHISM THEOREMS FOR RINGS 2 4 MAXIMAL AND PRIME IDEALS 2 5 POLYNOMIAL RINGS 2 6 UNIQUE FACTORIZATION 2 7 PRINCIPAL IDEAL DOMAINS AND EUCLIDEAN DOMAINS 2 8 RINGS OF FRACTIONS 2 9 IRREDUCIBLE POLYNOMIALS CHAPTER FIELD FUNDAMENTALS 3 1 FIELD EXTENSIONS 3 2 SPLITTING FIELDS 3 3 ALGEBRAIC CLOSURES 3 4 SEPARABILITY 3 5 NORMAL EXTENSIONS CHAPTER MODULE FUNDAMENTALS 4 1 MODULES AND ALGEBRAS 4 2 THE ISOMORPHISM THEOREMS FOR MODULES 4 3 DIRECT SUMS AND FREE MODULES 4 4 HOMOMORPHISMS AND MATRICES 4 5 SMITH NORMAL FORM 4 6 FUNDAMENTAL STRUCTURE THEOREMS 4 7 EXACT SEQUENCES AND DIAGRAM CHASING CHAPTER SOME BASIC TECHNIQUES OF GROUP THEORY 5 1 GROUPS ACTING ON SETS 5 2 THE ORBIT-STABILIZER THEOREM 5 3 APPLICATIONS TO COMBINATORICS 5 4 THE SYLOW THEOREMS 5 5 APPLICATIONS OF THE SYLOW THEOREMS 5 6 COMPOSITION SERIES 5 7 SOLVABLE AND NILPOTENT GROUPS 5 8 GENERATORS AND RELATIONS CHAPTER GALOIS THEORY 6 1 FIXED FIELDS AND GALOIS GROUPS 6 2 THE FUNDAMENTAL THEOREM 6 3 COMPUTING A GALOIS GROUP DIRECTLY 6 4 FINITE FIELDS 6 5 CYCLOTOMIC FIELDS 6 6 THE GALOIS GROUP OF A CUBIC 6 7 CYCLIC AND KUMMER EXTENSIONS 6 8 SOLVABILITY BY RADICALS 6 9 TRANSCENDENTAL EXTENSIONS APPENDIX TO CHAPTER CHAPTER INTRODUCING ALGEBRAIC NUMBER THEORY 7 1 INTEGRAL EXTENSIONS 7 2 QUADRATIC EXTENSIONS OF THE RATIONALS 7 3 NORMS AND TRACES 7 4 THE DISCRIMINANT 7 5 NOETHERIAN AND ARTINIAN MODULES AND RINGS 7 6 FRACTIONAL IDEALS 7 7 UNIQUE FACTORIZATION OF IDEALS IN A DEDEKIND DOMAIN 7 8 SOME ARITHMETIC IN DEDEKIND DOMAINS 7 9 P-ADIC NUMBERS CHAPTER INTRODUCING ALGEBRAIC GEOMETRY 8 1 VARIETIES 8 2 THE HILBERT BASIS THEOREM 8 3 THE NULLSTELLENSATZ: PRELIMINARIES 8 4 THE NULLSTELLENSATZ: EQUIVALENT VERSIONS AND PROOF 8 5 LOCALIZATION 8 6 PRIMARY DECOMPOSITION 8 7 TENSOR PRODUCT OF MODULES OVER A COMMUTATIVE RING 8 8 GENERAL TENSOR PRODUCTS CHAPTER INTRODUCING NONCOMMUTATIVE ALGEBRA 9 1 SEMISIMPLE MODULES 9 2 TWO KEY THEOREMS 9 3 SIMPLE AND SEMISIMPLE RINGS 9 4 FURTHER PROPERTIES OF SIMPLE RINGS, MATRIX RINGS, AND ENDOMORPHISMS 9 5 THE STRUCTURE OF SEMISIMPLE RINGS 9 6 MASCHKE’S THEOREM 9 7 THE JACOBSON RADICAL 9 8 THEOREMS OF HOPKINS-LEVITZKI AND NAKAYAMA CHAPTER INTRODUCING HOMOLOGICAL ALGEBRA 10 1 CATEGORIES 10 2 PRODUCTS AND COPRODUCTS 10 3 FUNCTORS 10 4 EXACT FUNCTORS 10 5 PROJECTIVE MODULES 10 6 INJECTIVE MODULES