The Devil You Know: 3 short stories - One from The Book of the Ancestor, two from The Red Queen's War

This book is intended as a basic text for a one-year course in Algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra. For the revised third edition, the author has added exercises and made numerous corrections to the text. Comments on Serge Lang's Algebra: Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books. April 1999 Notices of the AMS, announcing that the author was awarded the Leroy P. Steele Prize for Mathematical Exposition for his many mathematics books. The author has an impressive knack for presenting the important and interesting ideas of algebra in just the "right" way, and he never gets bogged down in the dry formalism which pervades some parts of algebra. MathSciNet's review of the first edition Algebra Front Cover Title Page Copyright Page Foreword Logical Prerequisites Contents Part 1: The Basic Objects of Algebra Chapter 1: Groups §1. Monoids §2. Groups §3. Normal subgroups §4. Cyclic groups §5. Operations of a group on a set §6. Sylow subgroups §7. Direct sums and free abelian groups §8. Finitely generated abelian groups §9. The dual group §10. Inverse limit and completion §11. Categories and functors §12. Free groups Exercises Semidirect product Some operations Explicit determination of groups Abelian groups Primitive groups Fiber products and coproducts pull-backs and push-outs Chapter 2: Rings §1. Rings and homomorphisms §2. Commutative rings §3. Polynomials and group rings §4. Localization §5. Principal and factorial rings Exercises Dedekind rings Chapter 3: Modules §1 . Basic definitions §2. The group of homomorphisms §3. Direct products and sums of modules §4. Free modules §5. Vector spaces §6. The dual space and dual module §7. Modules over principal rings §8. Euler-Poincaré maps §9. The snake lemma §10. Direct and inverse limits Exercises Localization Projective modules over Dedekind rings A few snakes Inverse limits Direct limits Graded Algebras Chapter 4: Polynomials §1. Basic properties for polynomials in one variable §2. Polynomials over a Factorial Ring §3. Criteria for irreducibility §4. Hilbert's theorem §5. Partial fractions §6. Symmetric polynomials §7. Mason-Stothers theorem and the abc conjecture §8. The resultant §9. Power series Exercises Integral-valued polynomials Exercises on symmetric functions λ-rings Part 2: Algebraic Equations Chapter 5: Algebraic Extensions §1. Finite and algebraic extensions §2. Algebraic closure §3. Splitting fields and normal extensions §4. Separable extensions §5. Finite fields §6. Inseparable extensions Exercises Chapter 6: Galois Theory §1. Galois extensions §2. Examples and applications §3. Roots of unity §4. Linear independence of characters §5. The norm and trace §6. Cyclic extensions §7. Solvable and radical extensions §8. Abelian Kummer theory §9. the Equation X^n-a=0 §10. Galois cohomology §11. Non-abelian Kummer extensions §12. Algebraic independence of homomorphisms §13. The normal basis theorem §14. Infinite Galois extensions §15. The modular connection Exercises Cyclotomic fields Rational functions Some aspects of Kummer theory Witt vectors Further Progress and directions Chapter 7: Extensions of Rings §1. Integral ring extensions §2. Integral Galois extensions §3. Extension of homomorphisms Exercises Symmetric Polynomials Chapter 8: Transcendental Extensions §1. Transcendence bases §2. Noether normalization theorem §3. Linearly disjoint extensions §4. Separable and regular extensions §5. Derivations Exercises Chapter 9: Algebraic Spaces §1. Hilbert's Nullstellensatz §2. Algebraic sets, spaces and varieties §3. Projections and elimination §4. Resultant systems §5. Spec of a ring Exercises Integrality Resultants Spec of a ring Chapter 10: Noetherian Rings and Modules §1. Basic criteria §2. Associated primes §3. Primary decomposition §4. Nakayama's lemma §5. Filtered and graded modules §6. The Hilbert polynomial §7. Indecomposable modules Exercises Locally constant dimensions Reduction of a complex mod p Comparison of homology at the special point Chapter 11: Real Fields §1. Ordered fields §2. Real fields §3. Real zeros and homomorphisms Exercises Real places Chapter 12: Absolute Values §1. Definitions, dependence, and independence §2. Completions §3. Finite extensions §4. Valuations §5. Completions and valuations §6. Discrete valuations §7. zeros of polynomials in complete fields Exercises Part 3: Linear Algebra and Representations Chapter 13: Matrices and Linear Maps §1. Matrices §2. The rank of a matrix §3. Matrices and linear maps §4. Determinants §5. Duality §6. Matrices and bilinear forms §7. Sesquilinear duality §8. The simplicity of SL_2(F)/±1 §9. The group SL_n(F), n≧3 Exercises Non-commutative cocycles Irreducibility of sI_n(f). Chapter 14: Representation of One Endomorphism §1. Representations §2. Decomposition over one endomorphism §3. The characteristic polynomial Exercises Diagonalizable endomorphisms Chapter 15: Structure of Bilinear Forms §1. Preliminaries, orthogonal sums §2. Quadratic maps §3. Symmetric forms, orthogonal bases §4. Symmetric forms over ordered fields §5. Hermitian forms §6. The spectral theorem (Hermitian case) §7. The spectral theorem (Symmetric case) §8. Alternating forms §9. The Pfaffian §10. Witt's theorem §11. The Witt group Exercises Symmetric endomorphisms Alternating forms The Witt group SL_n(R) Chapter 16: The Tensor Product §1. Tensor product §2. Basic properties §3. Flat modules §4. Extension of the base §5. Some functorial isomorphisms §6. Tensor product of algebras §7. The tensor algebra of a module §8. Symmetric products Exercises A little flatness Faithfully flat Tensor products and direct limits The Casimir element Chapter 17: Semisimplicity §1. Matrices and linear maps over non-commutative rings §2. Conditions defining semisimplicity §3. The density theorem §4. Semisimple rings §5. Simple rings §6. The Jacobson radical, base change, and tensor products §7. Balanced modules Exercises The radical Semisimple operations Chapter 18: Representations of Finite Groups §1. Representations and semisimplicity §2. Characters §3. 1-dimensional representations §4. The space of class functions §5. Orthogonality relations §6. Induced characters §7. Induced representations §8. Positive decomposition of the regular character §9. Supersolvable groups §10. Brauer's theorem §11. Field of definition of a representation §12. Example: GL_2 over a finite field Exercises Tensor product representations Chapter 19: The Alternating Product §1 Definition and basic properties §2. Fitting ideals §3. Universal derivations and the De Rham complex §4. The Clifford algebra Exercises Derivations Derivations and connections Some Clifford exercises Part 4: Homological Algebra Chapter 20: General Homology Theory §1. Complexes §2. Homology sequence §3. Euler characteristic and the Grothendieck group §4. Injective modules §5. Homotopies of morphisms of complexes §6. Derived functors §7. Delta-functors §8. Bifunctors §9. Spectral sequences Exercises Cohomology of groups Finite groups Injectives Tensor product of complexes Chapter 21: Finite Free Resolutions §1. Special complexes §2. Finite free resolutions §3. Unimodular polynomial vectors §4. The Koszul complex Exercises Appendix 1: The Transcendence of e and π Appendix 2: Some Set Theory §1. Denumerable Sets §2. Zorn's lemma §3. Cardinal mumbers §4. Well-ordering Exercises Bibliography Index Back Cover From April 1999 Notices of the AMS, announcing that the author was awarded the Leroy P. Steele Prize for Mathematical Exposition for his many mathematics books:'Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books.'From MathSciNet's review of the first edition:'The author has an impressive knack for presenting the important and interesting ideas of algebra in just the'right'way, and he never gets bogged down in the dry formalism which pervades some parts of algebra.'This book is intended as a basic text for a one-year course in Algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. This book successfully addresses all of the basic concepts of algebra. For the new edition, the author has added exercises and made numerous corrections to the text.

Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books. NOTICES OF THE AMS

The author has an impressive knack for presenting the important and interesting ideas of algebra in just the right way, and he never gets bogged down in the dry formalism which pervades some parts of algebra. MATHEMATICAL REVIEWS

This book is intended as a basic text for a one-year course in algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra. For the revised third edition, the author has added exercises and made numerous corrections to the text.

This book is intended as a basic text for a one year course in algebra at the graduate level or as a useful reference for mathematicians and professionals who use higher-level algebra.This book successfully addresses all of the basic concepts of algebra. For the new edition, the author has added exercises and made numerous corrections to the text.From MathSciNet's review of the first edition: "The author has an impressive knack for presenting the important and interesting ideas of algebra in just the "right" way, and he never gets bogged down in the dry formalism which pervades some parts of algebra." This book is intended as a basic text for a one year course in algebra at the graduate level or as a useful reference for mathematicians and professionals who use higher-level algebra. This book successfully addresses all of the basic concepts of algebra. For the new edition, the author has added exercises and made numerous corrections to the text. From MathSciNet's review of the first "The author has an impressive knack for presenting the important and interesting ideas of algebra in just the "right" way, and he never gets bogged down in the dry formalism which pervades some parts of algebra."