| The Basic Objects of Algebra / Part 1： |
| Groups / Chapter I： |
| Monoids / 1.： |
| Normal subgroups / 2.： |
| Cyclic groups / 4.： |
| Operations of a group on a set / 5.： |
| Sylow subgroups / 6.： |
| Direct sums and free abelian groups / 7.： |
| Finitely generated abelian groups / 8.： |
| The dual group / 9.： |
| Inverse limit and completion / 10.： |
| Categories and functors / 11.： |
| Free groups / 12.： |
| Rings / Chapter II： |
| Rings and homomorphisms |
| Commutative rings |
| Polynomials and group rings |
| Localization |
| Principal and factorial rings |
| Modules / Chapter III： |
| Basic definitions |
| The group of homomorphisms |
| Direct products and sums of modules |
| Free modules |
| Vector spaces |
| The dual space and dual module |
| Modules over principal rings |
| Euler-Poincare maps |
| The snake lemma |
| Direct and inverse limits |
| Polynomials / Chapter IV： |
| Basic properties for polynomials in one variable |
| Polynomials over a factorial ring |
| Criteria for irreducibility |
| Hilbert's theorem |
| Partial fractions |
| Symmetric polynomials |
| Mason-Stothers theorem and the abc conjecture |
| The resultant |
| Power series |
| Algebraic Equations / Part 2： |
| Algebraic Extensions / Chapter V： |
| Finite and algebraic extensions |
| Algebraic closure |
| Splitting fields and normal extensions |
| Separable extensions |
| Finite fields |
| Inseparable extensions |
| Galois Theory / Chapter VI： |
| Galois extensions |
| Examples and applications |
| Roots of unity |
| Linear independence of characters |
| The norm and trace |
| Cyclic extensions |
| Solvable and radical extensions |
| Abelian Kummer theory |
| The equation X[superscript n] - a = 0 |
| Galois cohomology |
| Non-abelian Kummer extensions |
| Algebraic independence of homomorphisms |
| The normal basis theorem / 13.： |
| Infinite Galois extensions / 14.： |
| The modular connection / 15.： |
| Extensions of Rings / Chapter VII： |
| Integral ring extensions |
| Integral Galois extensions |
| Extension of homomorphisms |
| Transcendental Extensions / Chapter VIII： |
| Transcendence bases |
| Noether normalization theorem |
| Linearly disjoint extensions |
| Separable and regular extensions |
| Derivations |
| Algebraic Spaces / Chapter IX： |
| Hilbert's Nullstellensatz |
| Algebraic sets, spaces and varieties |
| Projections and elimination |
| Resultant systems |
| Spec of a ring |
| Noetherian Rings and Modules / Chapter X： |
| Basic criteria |
| Associated primes |
| Primary decomposition |
| Nakayama's lemma |
| Filtered and graded modules |
| The Hilbert polynomial |
| Indecomposable modules |
| Real Fields / Chapter XI： |
| Ordered fields |
| Real fields |
| Real zeros and homomorphisms |
| Absolute Values / Chapter XII： |
| Definitions, dependence, and independence |
| Completions |
| Finite extensions |
| Valuations |
| Completions and valuations |
| Discrete valuations |
| Zeros of polynomials in complete fields |
| Linear Algebra and Representations / Part 3： |
| Matrices and Linear Maps / Chapter XIII： |
| Matrices |
| The rank of a matrix |
| Matrices and linear maps |
| Determinants |
| Duality |
| Matrices and bilinear forms |
| Sesquilinear duality |
| The simplicity of SL[subscript 2](F)/[plus or minus]1 |
| The group SL[subscript n](F), n [greater than or equal] 3 |
| Representation of One Endomorphism / Chapter XIV： |
| Representations |
| Decomposition over one endomorphism |
| The characteristic polynomial |
| Structure of Bilinear Forms / Chapter XV： |
| Preliminaries, orthogonal sums |
| Quadratic maps |
| Symmetric forms, orthogonal bases |
| Symmetric forms over ordered fields |
| Hermitian forms |
| The spectral theorem (hermitian case) |
| The spectral theorem (symmetric case) |
| Alternating forms |
| The Pfaffian |
| Witt's theorem |
| The Witt group |
| The Tensor Product / Chapter XVI： |
| Tensor product |
| Basic properties |
| Flat modules |
| Extension of the base |
| Some functorial isomorphisms |
| Tensor product of algebras |
| The tensor algebra of a module |
| Symmetric products |
| Semisimplicity / Chapter XVII： |
| Matrices and linear maps over non-commutative rings |
| Conditions defining semisimplicity |
| The density theorem |
| Semisimple rings |
| Simple rings |
| The Jacobson radical, base change, and tensor products |
| Balanced modules |
| Representations of Finite Groups / Chapter XVIII： |
| Representations and semisimplicity |
| Characters |
| 1-dimensional representations |
| The space of class functions |
| Orthogonality relations |
| Induced characters |
| Induced representations |
| Positive decomposition of the regular character |
| Supersolvable groups |
| Brauer's theorem |
| Field of definition of a representation |
| Example: GL[subscript 2] over a finite field |
| The Alternating Product / Chapter XIX： |
| Definition and basic properties |
| Fitting ideals |
| Universal derivations and the de Rham complex |
| The Clifford algebra |
| Homological Algebra / Part 4： |
| General Homology Theory / Chapter XX： |
| Complexes |
| Homology sequence |
| Euler characteristic and the Grothendieck group |
| Injective modules |
| Homotopies of morphisms of complexes |
| Derived functors |
| Delta-functors |
| Bifunctors |
| Spectral sequences |
| Finite Free Resolutions / Chapter XXI： |
| Special complexes |
| Finite free resolutions |
| Unimodular polynomial vectors |
| The Koszul complex |
| The Transcendence of e and [Pi] / Appendix 1： |
| Some Set Theory / Appendix 2： |
| Bibliography |
| Index |
| The Basic Objects of Algebra / Part 1： |
| Groups / Chapter I： |
| Monoids / 1.： |
| Normal subgroups / 2.： |
| Cyclic groups / 4.： |
| Operations of a group on a set / 5.： |
