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" Algebra / "
Serge Lang.
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BL
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| Record Number
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627702
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dltt
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| Main Entry
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Lang, Serge,1927-2005.
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| Title & Author
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Algebra /\ Serge Lang.
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| Edition Statement
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Rev. 3rd ed.
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| Publication Statement
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New York :: Springer,, c2002.
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| Series Statement
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Graduate texts in mathematics ;; 211
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| Page. NO
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xv, 914 p. ;; 24 cm.
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| ISBN
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038795385X (alk. paper)
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: 9780387953854 (alk. paper)
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Part 1 The Basic Objects of Algebra -- Chapter I Groups 3 -- 1. Monoids 3 -- 2. Groups 7 -- 3. Normal subgroups 13 -- 4. Cyclic groups 23 -- 5. Operations of a group on a set 25 -- 6. Sylow subgroups 33 -- 7. Direct sums and free abelian groups 36 -- 8. Finitely generated abelian groups 42 -- 9. The dual group 46 -- 10. Inverse limit and completion 49 -- 11. Categories and functors 53 -- 12. Free groups 66 -- Chapter II Rings 83 -- 1. Rings and homomorphisms 83 -- 2. Commutative rings 92 -- 3. Polynomials and group rings 97 -- 4. Localization 107 -- 5. Principal and factorial rings 111 -- Chapter III Modules 117 -- 2. The group of homomorphisms 122 -- 3. Direct products and sums of modules 127 -- 4. Free modules 135 -- 5. Vector spaces 139 -- 6. The dual space and dual module 142 -- 7. Modules over principal rings 146 -- 8. Euler-Poincare maps 155 -- 9. The snake lemma 157 -- 10. Direct and inverse limits 159 -- Chapter IV Polynomials 173 -- 1. Basic properties for polynomials in one variable 173 -- 2. Polynomials over a factorial ring 180 -- 3. Criteria for irreducibility 183 -- 4. Hilbert's theorem 186 -- 5. Partial fractions 187 -- 6. Symmetric polynomials 190 -- 7. Mason-Stothers theorem and the abc conjecture 194 -- 8. The resultant 199 -- 9. Power series 205 -- Part 2 Algebraic Equations -- Chapter V Algebraic Extensions 223 -- 1. Finite and algebraic extensions 225 -- 2. Algebraic closure 229 -- 3. Splitting fields and normal extensions 236 -- 4. Separable extensions 239 -- 5. Finite fields 244 -- 6. Inseparable extensions 247 -- Chapter VI Galois Theory 261 -- 1. Galois extensions 261 -- 2. Examples and applications 269 -- 3. Roots of unity 276 -- 4. Linear independence of characters 282 -- 5. The norm and trace 284 -- 6. Cyclic extensions 288 -- 7. Solvable and radical extensions 291 -- 8. Abelian Kummer theory 293 -- 9. The equation X[superscript n] -- a = 0 297 -- 10. Galois cohomology 302 -- 11. Non-abelian Kummer extensions 304 -- 12. Algebraic independence of homomorphisms 308 -- 13. The normal basis theorem 312 -- 14. Infinite Galois extensions 313 -- 15. The modular connection 315 -- Chapter VII Extensions of Rings 333 -- 1. Integral ring extensions 333 -- 2. Integral Galois extensions 340 -- 3. Extension of homomorphisms 346 -- Chapter VIII Transcendental Extensions 355 -- 1. Transcendence bases 355 -- 2. Noether normalization theorem 357 -- 3. Linearly disjoint extensions 360 -- 4. Separable and regular extensions 363 -- 5. Derivations 368 -- Chapter IX Algebraic Spaces 377 -- 1. Hilbert's Nullstellensatz 377 -- 2. Algebraic sets, spaces and varieties 381 -- 3. Projections and elimination 388 -- 4. Resultant systems 401 -- 5. Spec of a ring 405 -- Chapter X Noetherian Rings and Modules 413 -- 1. Basic criteria 413 -- 2. Associated primes 416 -- 3. Primary decomposition 421 -- 4. Nakayama's lemma 424 -- 5. Filtered and graded modules 426 -- 6. The Hilbert polynomial 431 -- 7. Indecomposable modules 439 -- Chapter XI Real Fields 449 -- 1. Ordered fields 449 -- 2. Real fields 451 -- 3. Real zeros and homomorphisms 457 -- Chapter XII Absolute Values 465 -- 1. Definitions, dependence, and independence 465 -- 2. Completions 468 -- 3. Finite extensions 476 -- 4. Valuations 480 -- 5. Completions and valuations 486 -- 6. Discrete valuations 487 -- 7. Zeros of polynomials in complete fields 491 -- Part 3 Linear Algebra and Representations -- Chapter XIII Matrices and Linear Maps 503 -- 1. Matrices 503 -- 2. The rank of a matrix 506 -- 3. Matrices and linear maps 507 -- 4. Determinants 511 -- 5. Duality 522 -- 6. Matrices and bilinear forms 527 -- 7. Sesquilinear duality 531 -- 8. The simplicity of SL[subscript 2](F)/[plus or minus]1 536 -- 9. The group SL[subscript n](F), n [greater than or equal] 3 540 -- Chapter XIV Representation of One Endomorphism 553 -- 1. Representations 553 -- 2. Decomposition over one endomorphism 556 -- 3. The characteristic polynomial 561 -- Chapter XV Structure of Bilinear Forms 571 -- 1. Preliminaries, orthogonal sums 571 -- 2. Quadratic maps 574 -- 3. Symmetric forms, orthogonal bases 575 -- 4. Symmetric forms over ordered fields 577 -- 5. Hermitian forms 579 -- 6. The spectral theorem (hermitian case) 581 -- 7. The spectral theorem (symmetric case) 584 -- 8. Alternating forms 586 -- 9. The Pfaffian 588 -- 10. Witt's theorem 589 -- 11. The Witt group 594 -- Chapter XVI The Tensor Product 601 -- 1. Tensor product 601 -- 2. Basic properties 607 -- 3. Flat modules 612 -- 4. Extension of the base 623 -- 5. Some functorial isomorphisms 625 -- 6. Tensor product of algebras 629 -- 7.
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| Subject
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Algebra.
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| LC Classification
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QA154.3.L3 2002
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