.

0 (hodnocen0 x )

(5.5) Půjčeno:11x 

BK

Revised 3rd ed.

New York : Springer, c2002

xv,914 s.

ISBN 0-387-95385-X (váz.)

Graduate texts in mathematics ; 211

Obsahuje předmluvu, dodatky, rejstřík

Bibliografie: s. 895-901

Algebra - učebnice vysokošk.

000002713

Part One - The Basic Objects of Algebra // Chapter I Groups // 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 // 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 // 1. Basic definitions 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-Poincaré maps 155 // 9. The snake lemma 157 // 10. Direct and inverse limits 159 // Chapter IV Polynomials // 1. Basic properties for polynomials in one variable // 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 // 8. The resultant 199 // 9. Power series 205 // Part Two Algebraic Equations // Chapter V Algebraic Extensions // 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 // 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" - 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 // 1. Integral ring extensions 333 // 2. Integral Galois extensions 340 // 3. Extension of homomorphisms 346 // 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 // 1. Hilbert’s Nullstellensatz 378 // 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 Three Linear Algebra and Representations // Chapter IX Algebraic Spaces // 377 // Chapter XIII Matrices and Linear Maps // 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 SL2(F)/ ± 1 536 // 9. The group SLn(F), n  3 540 // Chapter XIV Representation of One Endomorphism // 1. Representations 553 // 2. Decomposition over one endomorphism 556 // 3. The characteristic polynomial 561 //

Chapter XV Structure of Bilinear Forms // 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 // 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. The tensor algebra of a module 632 // 8. Symmetric products 635 // Chapter XVII Semisimplicity // 1. Matrices and linear maps over non-commutative rings 641 // 2. Conditions defining semisimplicity 645 // 3. The density theorem 646 // 4. Semisimple rings 651 // 5. Simple rings 654 // 6. The Jacobson radical, base change, and tensor products 657 // 7. Balanced modules 660 // Chapter XVIII Representations of Finite Groups // 1. Representations and semisimplicity 663 // 2. Characters 667 3 i-dimensional representations // 4. The space of class functions // 5. Orthogonality relations // 6. Induced characters 686 // 7. Induced representations 688 8 Positive decomposition of the regular character // 9. Supersolvable groups 702 // 10. Brauer’s theorem 704 // 11. Field of definition of a representation 710 // 12. Example: GL, over a finite field 712 // Chapter XIX The Alternating Product // 1. Definition and basic properties 731 // 2. Fitting ideals 738 // 3. Universal derivations and the de Rham complex 746 // 4. The Clifford algebra 749 // Part Four Homological Algebra // Chapter XX General Homology Theory 761 // 1. Complexes 761 // 2. Homology sequence 767 // 3. Euler characteristic and the Grothendieck group 769 //

4. Injective modules 782 // 5. Homotopies of morphisms of complexes 787 // 6. Derived functors 790 // 7. Delta-functors 799 // 8. Bifunctors 806 // 9. Spectral sequences 814 // Chapter XXI Finite Free Resolutions 835 // 1. Special complexes 835 // 2. Finite free resolutions 839 // 3. Unimodular polynomial vectors 846 // 4. The Koszul complex 850 // Appendix 1 The Transcendence of e and n 867 // Appendix 2 Some Set Theory 875 // Bibliography 895 // index 903