ADVANCED MODERN ALGEBRA: THIRD EDITION, PART 1.VOLUME: 165

This new edition, now in two parts, has been significantly reorganized and many sections have been rewritten. This first part, designed for a first year of graduate algebra, consists of two courses: Galois theory and Module theory. Topics covered in the first course are classical formulas for solutions of cubic and quartic equations, classical number theory, commutative algebra, groups, and Galois theory. Topics in the second course are Zorn's lemma, canonical forms, inner product spaces, categories and limits, tensor products, projective, injective, and flat modules, multilinear algebra, affine varieties, and Gröbner bases.

Readership
Graduate students and researchers interested in learning and teaching algebra.

Author
Joseph J. Rotman, University of Illinois at Urbana-Champaign, IL.

Contents
Preface to Third Edition: Part 1 xi
Acknowledgments xiv
Part A. Course I
Chapter A-1. Classical Formulas 3
Cubics 4
Quartics 6
Chapter A-2. Classical Number Theory 9
Divisibility 9
Euclidean Algorithms 16
Congruence 19
Chapter A-3. Commutative Rings 29
Polynomials 41
Homomorphisms 47
Quotient Rings 55
From Arithmetic to Polynomials 62
Maximal Ideals and Prime Ideals 74
Finite Fields 83
Irreducibility 89
Euclidean Rings and Principal Ideal Domains 97
Unique Factorization Domains 104
Chapter A-4. Groups 115
Permutations 116
Even and Odd 123
Groups 127
Lagrange’s Theorem 139
Homomorphisms 150
Quotient Groups 159
Simple Groups 173
Chapter A-5. Galois Theory 179
Insolvability of the Quintic 179
Classical Formulas and Solvability by Radicals 187
Translation into Group Theory 190
Fundamental Theorem of Galois Theory 200
Calculations of Galois Groups 223
Chapter A-6. Appendix: Set Theory 235
Equivalence Relations 243
Chapter A-7. Appendix: Linear Algebra 247
Vector Spaces 247
Linear Transformations and Matrices 259
Part B. Course II
Chapter B-1. Modules 273
Noncommutative Rings 273
Chain Conditions on Rings 282
Left and Right Modules 288
Chain Conditions on Modules 300
Exact Sequences 305
Chapter B-2. Zorn’s Lemma 313
Zorn, Choice, and Well-Ordering 313
Zorn and Linear Algebra 319
Zorn and Free Abelian Groups 323
Semisimple Modules and Rings 334
Algebraic Closure 339
Transcendence 345
L¨uroth’s Theorem 353
Chapter B-3. Advanced Linear Algebra 359
Torsion and Torsion-free 359
Basis Theorem 362
Fundamental Theorem 371
Elementary Divisors 371
Invariant Factors 374
From Abelian Groups to Modules 378
Rational Canonical Forms 383
Eigenvalues 388
Jordan Canonical Forms 395
Smith Normal Forms 402
Inner Product Spaces 417
Orthogonal and Symplectic Groups 429
Hermitian Forms and Unitary Groups 436
Chapter B-4. Categories of Modules 441
Categories 441
Functors 461
Galois Theory for Infinite Extensions 475
Free and Projective Modules 481
Injective Modules 492
Divisible Abelian Groups 501
Tensor Products 509
Adjoint Isomorphisms 522
Flat Modules 529
Chapter B-5. Multilinear Algebra 543
Algebras and Graded Algebras 543
Tensor Algebra 552
Exterior Algebra 561
Grassmann Algebras 566
Exterior Algebra and Differential Forms 573
Determinants 575
Chapter B-6. Commutative Algebra II 591
Old-Fashioned Algebraic Geometry 591
Affine Varieties and Ideals 593
Nullstellensatz 599
Nullstellensatz Redux 604
Irreducible Varieties 614
Affine Morphisms 623
Algorithms in k[x1,...,xn] 628
Monomial Orders 629
Division Algorithm 636
Gr¨obner Bases 639
Chapter B-7. Appendix: Categorical Limits 651
Inverse Limits 651
Direct Limits 657
Directed Index Sets 659
Adjoint Functors 666
Chapter B-8. Appendix: Topological Spaces 673
Topological Groups 678
Bibliography 681
Special Notation 687
Index 693