copyright page -- title page -- Contents -- Preface -- A Guide to this Guide -- Algebra: Classical, Modern, and Ultramodern -- The Beginnings of Modern Algebra -- Modern Algebra -- Ultramodern Algebra -- What Next? -- Categories -- Categories -- Functors -- Natural Transformations -- Products, Coproducts, and Generalizations -- Algebraic Structures -- Structures with One Operation -- Rings -- Actions -- Semirings -- Algebras -- Ordered Structures -- Groups and their Representations -- Definitions -- Groups and homomorphisms -- Subgroups -- Actions -- G acting on itself -- Some Important Examples -- Permutation groups -- Symmetry groups -- Other examples -- Topological groups -- Free groups -- Reframing the Definitions -- Orbits and Stabilizers -- Stabilizers -- Orbits -- Acting by multiplication -- Cosets -- Counting cosets and elements -- Double cosets -- A nice example -- Homomorphisms and Subgroups -- Kernel, image, quotient -- Homomorphism theorems -- Exact sequences -- Hölder's dream -- Many Cheerful Subgroups -- Generators, cyclic groups -- Elements of finite order -- Finitely generated groups and the Burnside problem -- Other nice subgroups -- Conjugation and the class equation -- p-groups -- Sylow's Theorem and Sylow subgroups -- Sequences of Subgroups -- Composition series -- Central series, derived series, nilpotent, solvable -- New Groups from Old -- Direct products -- Semidirect products -- Isometries of R3 -- Free products -- Direct sums of abelian groups -- Inverse limits and direct limits -- Generators and Relations -- Definition and examples -- Cayley graphs -- The word problem -- Abelian Groups -- Torsion -- The structure theorem -- Small Groups -- Order four, order p2 -- Order six, order pq -- Order eight, order p3 -- And so on -- Groups of Permutations -- Cycle notation and cycle structure -- Conjugation and cycle structure.

Transpositions as generators -- Signs and the alternating groups -- Transitive subgroups -- Automorphism group of S_n -- Some Linear Groups -- Definitions and examples -- Generators -- The regular representation -- Diagonal and upper triangular -- Normal subgroups -- PGL -- Linear groups over finite fields -- Representations of Finite Groups -- Definitions -- Examples -- Constructions -- Decomposing into irreducibles -- Direct products -- Characters -- Character tables -- Going through quotients -- Going up and down -- Representations of S_4 -- Rings and Modules -- Definitions -- Rings -- Modules -- Torsion -- Bimodules -- Ideals -- Restriction of scalars -- Rings with few ideals -- More Examples, More Definitions -- The integers -- Fields and skew fields -- Polynomials -- R-algebras -- Matrix rings -- Group algebras -- Monsters -- Some examples of modules -- Nil and nilpotent ideals -- Vector spaces as K[X]-modules -- Q and Q/Z as Z-modules -- Why study modules? -- Homomorphisms, Submodules, and Ideals -- Submodules and quotients -- Quotient rings -- Irreducible modules, simple rings -- Small and large submodules -- Composing and Decomposing -- Direct sums and products -- Complements -- Direct and inverse limits -- Products of rings -- Free Modules -- Definitions and examples -- Vector spaces -- Traps -- Generators and free modules -- Homomorphisms of free modules -- Invariant basis number -- Commutative Rings, Take One -- Prime ideals -- Primary ideals -- The Chinese Remainder Theorem -- Rings of Polynomials -- Degree -- The evaluation homomorphism -- Integrality -- Unique factorization and ideals -- Derivatives -- Symmetric polynomials and functions -- Polynomials as functions -- The Radical, Local Rings, and Nakayama's Lemma -- The Jacobson radical -- Local rings -- Nakayama's Lemma -- Commutative Rings: Localization -- Localization.

Fields of fractions -- An important example -- Modules under localization -- Ideals under localization -- Integrality under localization -- Localization at a prime ideal -- What if R is not commutative? -- Hom -- Making Hom a module -- Functoriality -- Additivity -- Exactness -- Tensor Products -- Definition and examples -- Examples -- Additivity and exactness -- Over which ring? -- When R is commutative -- Extension of scalars, aka base change -- Extension and restriction -- Tensor products and Hom -- Finite free modules -- Tensoring a module with itself -- Tensoring two rings -- Projective, Injective, Flat -- Projective modules -- Injective modules -- Flat modules -- If R is commutative -- Finiteness Conditions for Modules -- Finitely generated and finitely cogenerated -- Artinian and Noetherian -- Finite length -- Semisimple Modules -- Definitions -- Basic properties -- Socle and radical -- Finiteness conditions -- Structure of Rings -- Finiteness conditions for rings -- Simple Artinian rings -- Semisimple rings -- Artinian rings -- Non-Artinian rings -- Factorization in Domains -- Units, irreducibles, and the rest -- Existence of factorization -- Uniqueness of factorization -- Principal ideal domains -- Euclidean domains -- Greatest common divisor -- Dedekind domains -- Finitely Generated Modules over Dedekind Domains -- The structure theorems -- Application to abelian groups -- Application to linear transformations -- Fields and Skew Fields -- Fields and Algebras -- Some examples -- Characteristic and prime fields -- K-algebras and extensions -- Two kinds of K-homomorphisms -- Generating sets -- Compositum -- Linear disjointness -- Algebraic Extensions -- Definitions -- Transitivity -- Working without an A -- Norm and trace -- Algebraic elements and homomorphisms -- Splitting fields -- Algebraic closure -- Finite Fields.

Transcendental Extensions -- Transcendence basis -- Geometric examples -- Noether Normalization -- Luroth's Theorem -- Symmetric functions -- Separability -- Separable and inseparable polynomials -- Separable extensions -- Separability and tensor products -- Norm and trace -- Purely inseparable extensions -- Separable closure -- Primitive elements -- Automorphisms and Normal Extensions -- Automorphisms -- Normal extensions -- Galois Theory -- Galois extensions and Galois groups -- The Galois group as topological group -- The Galois correspondence -- Composita -- Norm and trace -- Normal bases -- Solution by radicals -- Determining Galois groups -- The inverse Galois problem -- Analogies and generalizations -- Skew Fields and Central Simple Algebras -- Definition and basic results -- Quaternion algebras -- Skew fields over R -- Tensor products -- Splitting fields -- Reduced norms and traces -- The Skolem-Noether Theorem -- The Brauer group -- Bibliography -- Index of Notations -- Index -- About the Author.