| preface . etymology special notation chapter 1 things past 1.1. some number theory 1.2. roots of unity 1.3. some set theory chapter 2 groups i 2.1. introduction 2.2. permutations 2.3. groups 2.4. lagrange's theorem 2.5. homomorphisms 2.6. quotient groups 2.7. group actions chapter 3 commutative rings i 3.1. introduction 3.2. first properties 3.3. polynomials 3.4. greatest common divisors .3.5. homomorphisms 3.6. euclidean rings 3.7. linear algebra vector spaces linear transformations 3.8. quotient rings and finite fields chapter 4 fields 4.1. insolvability of the quintic formulas and solvability by radicals translation into group theory 4.2. fundamental theorem of galois theory chapter 5 groups ii 5.1. finite abelian groups direct sums basis theorem fundamental theorem 5.2. the sylow theorems 5.3. the jordan-h61der theorem 5.4. projective unimodular groups 5.5. presentations 5.6. the neilsen-schreier theorem chapter 6 commutative rings ii 6.1. prime ideals and maximal ideals 6.2. unique factorization domains 6.3. noetherian rings 6.4. applications of zorn's lemma 6.5. varieties 6.6. gr6bner bases .. generalized division algorithm buchberger's algorithm chapter 7 modules and categories 7.1. modules 7.2. categories 7.3. functors 7.4. free modules, projectives, and injectives 7.5. grothendieck groups 7.6. limits chapter 8 algebras 8.1. noncommutative rings 8.2. chain conditions 8.3. semisimple rings 8.4. tensor products 8.5. characters 8.6. theorems of burnside and of frobenius chapter 9 advanced linear algebra 9.1. modules over pids 9.2. rational canonical forms 9.3. jordan canonical forms 9.4. smith normal forms 9.5. bilinear forms 9.6, graded algebras 9.7. division algebras 9.8. exterior algebra 9.9. determinants 9.10. lie algebras chapter 10 homology 10.1. introduction 10.2. semidirect products 10.3. general extensions and cohomology 10.4. homology functors 10.5. derived functors 10.6. ext and tot 10.7. cohomology of groups 10.8. crossed products 10.9. introduction to spectral sequences chapter 11 commutative rings iii 11.1. local and global 11.2. dedekind rings integrality nullstellensatz redux algebraic integers characterizations of dedekind rings finitely generated modules over dedekind rings 11.3. global dimension 11.4. regular local rings appendix the axiom of choice and zorn's lemma bibliography index ... |
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