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| The Basic Objects of Algebra | |
| Groups | p. 3 |
| Monoids | p. 3 |
| Groups | p. 7 |
| Normal subgroups | p. 13 |
| Cyclic groups | p. 23 |
| Operations of a group on a set | p. 25 |
| Sylow subgroups | p. 33 |
| Direct sums and free abelian groups | p. 36 |
| Finitely generated abelian groups | p. 42 |
| The dual group | p. 46 |
| Inverse limit and completion | p. 49 |
| Categories and functors | p. 53 |
| Free groups | p. 66 |
| Rings | p. 83 |
| Rings and homomorphisms | p. 83 |
| Commutative rings | p. 92 |
| Polynomials and group rings | p. 97 |
| Localization | p. 107 |
| Principal and factorial rings | p. 111 |
| Modules | p. 117 |
| Basic definitions | p. 117 |
| The group of homomorphisms | p. 122 |
| Direct products and sums of modules | p. 127 |
| Free modules | p. 135 |
| Vector spaces | p. 139 |
| The dual space and dual module | p. 142 |
| Modules over principal rings | p. 146 |
| Euler-Poincare maps | p. 155 |
| The snake lemma | p. 157 |
| Direct and inverse limits | p. 159 |
| Polynomials | p. 173 |
| Basic properties for polynomials in one variable | p. 173 |
| Polynomials over a factorial ring | p. 180 |
| Criteria for irreducibility | p. 183 |
| Hilbert's theorem | p. 186 |
| Partial fractions | p. 187 |
| Symmetric polynomials | p. 190 |
| Mason-Stothers theorem and the abc conjecture | p. 194 |
| The resultant | p. 199 |
| Power series | p. 205 |
| Algebraic Equations | |
| Algebraic Extensions | p. 223 |
| Finite and algebraic extensions | p. 225 |
| Algebraic closure | p. 229 |
| Splitting fields and normal extensions | p. 236 |
| Separable extensions | p. 239 |
| Finite fields | p. 244 |
| Inseparable extensions | p. 247 |
| Galois Theory | p. 261 |
| Galois extensions | p. 261 |
| Examples and applications | p. 269 |
| Roots of unity | p. 276 |
| Linear independence of characters | p. 282 |
| The norm and trace | p. 284 |
| Cyclic extensions | p. 288 |
| Solvable and radical extensions | p. 291 |
| Abelian Kummer theory | p. 293 |
| The equation X[superscript n] - a = 0 | p. 297 |
| Galois cohomology | p. 302 |
| Non-abelian Kummer extensions | p. 304 |
| Algebraic independence of homomorphisms | p. 308 |
| The normal basis theorem | p. 312 |
| Infinite Galois extensions | p. 313 |
| The modular connection | p. 315 |
| Extensions of Rings | p. 333 |
| Integral ring extensions | p. 333 |
| Integral Galois extensions | p. 340 |
| Extension of homomorphisms | p. 346 |
| Transcendental Extensions | p. 355 |
| Transcendence bases | p. 355 |
| Noether normalization theorem | p. 357 |
| Linearly disjoint extensions | p. 360 |
| Separable and regular extensions | p. 363 |
| Derivations | p. 368 |
| Algebraic Spaces | p. 377 |
| Hilbert's Nullstellensatz | p. 377 |
| Algebraic sets, spaces and varieties | p. 381 |
| Projections and elimination | p. 388 |
| Resultant systems | p. 401 |
| Spec of a ring | p. 405 |
| Noetherian Rings and Modules | p. 413 |
| Basic criteria | p. 413 |
| Associated primes | p. 416 |
| Primary decomposition | p. 421 |
| Nakayama's lemma | p. 424 |
| Filtered and graded modules | p. 426 |
| The Hilbert polynomial | p. 431 |
| Indecomposable modules | p. 439 |
| Real Fields | p. 449 |
| Ordered fields | p. 449 |
| Real fields | p. 451 |
| Real zeros and homomorphisms | p. 457 |
| Absolute Values | p. 465 |
| Definitions, dependence, and independence | p. 465 |
| Completions | p. 468 |
| Finite extensions | p. 476 |
| Valuations | p. 480 |
| Completions and valuations | p. 486 |
| Discrete valuations | p. 487 |
| Zeros of polynomials in complete fields | p. 491 |
| Linear Algebra and Representations | |
| Matrices and Linear Maps | p. 503 |
| Matrices | p. 503 |
| The rank of a matrix | p. 506 |
| Matrices and linear maps | p. 507 |
| Determinants | p. 511 |
| Duality | p. 522 |
| Matrices and bilinear forms | p. 527 |
| Sesquilinear duality | p. 531 |
| The simplicity of SL[subscript 2](F)/[plus or minus]1 | p. 536 |
| The group SL[subscript n](F), n [greater than or equal] 3 | p. 540 |
| Representation of One Endomorphism | p. 553 |
| Representations | p. 553 |
| Decomposition over one endomorphism | p. 556 |
| The characteristic polynomial | p. 561 |
| Structure of Bilinear Forms | p. 571 |
| Preliminaries, orthogonal sums | p. 571 |
| Quadratic maps | p. 574 |
| Symmetric forms, orthogonal bases | p. 575 |
| Symmetric forms over ordered fields | p. 577 |
| Hermitian forms | p. 579 |
| The spectral theorem (hermitian case) | p. 581 |
| The spectral theorem (symmetric case) | p. 584 |
| Alternating forms | p. 586 |
| The Pfaffian | p. 588 |
| Witt's theorem | p. 589 |
| The Witt group | p. 594 |
| The Tensor Product | p. 601 |
| Tensor product | p. 601 |
| Basic properties | p. 607 |
| Flat modules | p. 612 |
| Extension of the base | p. 623 |
| Some functorial isomorphisms | p. 625 |
| Tensor product of algebras | p. 629 |
| The tensor algebra of a module | p. 632 |
| Symmetric products | p. 635 |
| Semisimplicity | p. 641 |
| Matrices and linear maps over non-commutative rings | p. 641 |
| Conditions defining semisimplicity | p. 645 |
| The density theorem | p. 646 |
| Semisimple rings | p. 651 |
| Simple rings | p. 654 |
| The Jacobson radical, base change, and tensor products | p. 657 |
| Balanced modules | p. 660 |
| Representations of Finite Groups | p. 663 |
| Representations and semisimplicity | p. 663 |
| Characters | p. 667 |
| 1-dimensional representations | p. 671 |
| The space of class functions | p. 673 |
| Orthogonality relations | p. 677 |
| Induced characters | p. 686 |
| Induced representations | p. 688 |
| Positive decomposition of the regular character | p. 699 |
| Supersolvable groups | p. 702 |
| Brauer's theorem | p. 704 |
| Field of definition of a representation | p. 710 |
| Example: GL[subscript 2] over a finite field | p. 712 |
| The Alternating Product | p. 731 |
| Definition and basic properties | p. 731 |
| Fitting ideals | p. 738 |
| Universal derivations and the de Rham complex | p. 746 |
| The Clifford algebra | p. 749 |
| Homological Algebra | |
| General Homology Theory | p. 761 |
| Complexes | p. 761 |
| Homology sequence | p. 767 |
| Euler characteristic and the Grothendieck group | p. 769 |
| Injective modules | p. 782 |
| Homotopies of morphisms of complexes | p. 787 |
| Derived functors | p. 790 |
| Delta-functors | p. 799 |
| Bifunctors | p. 806 |
| Spectral sequences | p. 814 |
| Finite Free Resolutions | p. 835 |
| Special complexes | p. 835 |
| Finite free resolutions | p. 839 |
| Unimodular polynomial vectors | p. 846 |
| The Koszul complex | p. 850 |
| The Transcendence of e and [Pi] | p. 867 |
| Some Set Theory | p. 875 |
| Bibliography | p. 895 |
| Index | p. 903 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
ISBN: 9780387953854
ISBN-10: 038795385X
Series: Graduate Texts in Mathematics
Published: 3rd August 2002
Format: Hardcover
Language: English
Number of Pages: 914
Audience: Professional and Scholarly
Publisher: Springer-Verlag New York Inc.
Country of Publication: GB
Edition Number: 3
Edition Type: Revised
Dimensions (cm): 24.1 x 16.3 x 4.7
Weight (kg): 1.47
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