Algebra / Serge Lang.
Editor: Menlo Park, Calif. : Addison-Wesley, Advanced Book Program, 1984Edición: 2nd edDescripción: xv, 714 p. ; 25 cmISBN: 0201054876Tema(s): AlgebraOtra clasificación: 00A05 (13-01 15-01 16-01 20-01)Part One Groups, Rings, and Modules Chapter I Groups [3] I Monoids [3] 2. Groups [7] 3. Cyclic groups [11] 4. Normal subgroups [12] 5. Operations of a group on a set [20] 6. Sylow subgroups [24] 7. Categories and functors [26] 8. Free groups [34] 9. Direct sums and free abelian groups [41] 10. Finitely generated abelian groups [47] 11. The dual group [51] Chapter II Rings [60] 1. Rings and homomorphisms [60] 2. Commutative rings [66] 3. Localization [71] 4. Principal rings [74] 5. Spec of a ring [77] Chapter III Modules [81] 1. Basic definitions [81] 2. The group of homomorphisms [84] 3. Direct products and sums of modules [86] 4. Free modules [92] 5. Vector spaces [93] 6. The dual space [96] 7. The snake lemma [100] 8. Projective and injective modules [101] 9. Direct and inverse limits [106] Chapter IV Homology [116] 1. Complexes [117] 2. Homology sequence [121] 3. Euler characteristic [123] 4. Special complexes [133] 5. Homotopies of morphisms of complexes [137] 6. Derived functors [141] 7. Delta-functors [148] 8. Bifunctors [155] 9. Spectral sequences [163] Chapter V Polynomials [176] 1. Free algebras [176] 2. Definition of polynomials [180] 3. Elementary properties of polynomials [185] 4. The Euclidean algorithm [190] 5. Partial fractions [194] 6. Unique factorization in several variables [197] 7. Criteria for irreducibility [200] 8. The derivative and multiple roots [202] 9. Symmetric polynomials [204] 10. The resultant [206] 11. Power series [211] Chapter VI Noetherian Rings and Modules [222] 1. Basic criteria [222] 2. Hilbert’s theorem [226] 3. Power series are Noetherian [227] 4. Associated primes [228] 5. Primary decomposition [233] 6. Nakayama’s lemma [236] 7. Filtered and graded modules [238] 8. The Hilbert polynomial [243] 9. Indecomposable modules [246] 10. Finite free resolutions [250] Part Two Field Theory Chapter VII Algebraic Extensions [265] 1. Finite and algebraic extensions [265] 2. Algebraic closure [271] 3. Splitting fields and normal extensions [278] 4. Separable extensions [281] 5. Finite fields [287] 6. Primitive elements [290] 7. Purely inseparable extensions [291] Chapter VIII Galois Theory [300] 1. Galois extensions [300] 2. Examples and applications [308] 3. Roots of unity [313] 4. Linear independence of characters [318] 5. The norm and trace [320] 6. Cyclic extensions [323] 7. Solvable and radical extensions [326] 8. Abelian Kummer theory [328] 9. The equation Xn- a = 0 [331] 10. Galois cohomology [334] 11. Non-abelian Kummer extensions [336] 12. Algebraic independence of homomorphisms [340] 13. The normal basis theorem [344] Chapter IX Extensions of Rings [355] 1. Integral ring extensions [355] 2. Integral Galois extensions [362] 3. Extension of homomorphisms [368] Chapter X Transcendental Extensions [372] 1. Transcendence bases [372] 2. Hilbert’s Nullstellensatz [374] 3. Algebraic sets [376] 4. Noether normalization theorem [378] 5. Linearly disjoint extensions [379] 6. Separable extensions [382] 7. Derivations [385] Chapter XI Real Fields [390] 1. Ordered fields [390] 2. Real fields [392] 3. Real zeros and homomorphisms [398] Chapter XII Absolute Values [404] 1. Definitions, dependence, and independence [404] 2. Completions [407] 3. Finite extensions [414] 4. Valuations [417] 5. Completions and valuations [425] 6. Discrete valuations [426] 7. Zeros of polynomials in complete fields [430] Part Three Linear Algebra and Representations Chapter XIII Matrices and Linear Maps [441] 1. Matrices [441] 2. The rank of a matrix [444] 3. Matrices and linear maps [445] 4. Determinants [449] 5. Duality [458] 6. Matrices and bilinear forms [463] 7. Sesquilinear duality [467] 8. The simplicity of SL2(F)/ ± 1 [472] 9. The group SLn(F), n >=3 [476] 10. Fitting ideals [480] 11. Unimodular polynomial vectors [488] Chapter XIV Structure of Bilinear Forms [498] 1. Preliminaries, orthogonal sums [498] 2. Quadratic maps [501] 3. Symmetric forms, orthogonal bases [502] 4. Hyperbolic spaces [503] 5. Witt’s theorem [505] 6. The Witt group [508] 7. Symmetric forms over ordered fields [509] 8. The Clifford algebra [511] 9. Alternating forms [515] 10. The Pfaffian [517] 11. Hermitian forms [519] 12. The spectral theorem (hermitian case) [521] 13. The spectral theorem (symmetric case) [524] Chapter XV Representation of One Endomorphism [529] 1. Representations [529] 2. Modules over principal rings [532] 3. Decomposition over one endomorphism [541] 4. The characteristic polynomial [545] Chapter XVI Multilinear Products [554] 1. Tensor product [554] 2. Basic properties [560] 3. Flat modules [565] 4. Extension of the base [575] 5. Some functorial isomorphisms 6. Tensor product of algebras [581] 7. The tensor algebra of a module [583] 8. Symmetric products [586] 9. Alternating products [588] 10. The Koszul complex [593] 11. The Grothendieck ring [605] 12. Universal derivations [610] Chapter XVII Semisimplicity [621] 1. Matrices and linear maps over non-commutative rings [621] 2. Conditions defining semisimplicity [625] 3. The density theorem [626] 4. Semisimple rings [629] 5. Simple rings [632] 6. Balanced modules [636] Chapter XVIII Representations of Finite Groups [639] 1. Semisimplicity of the group algebra [639] 2. Characters [641] 3. 1-dimensional representations [645] 4. The space of class functions [647] 5. Orthogonality relations [651] 6. Induced characters [659] 7. Induced representations [661] 8. Positive decomposition of the regular character [666] 9. Super solvable groups [668] 10. Brauer’s theorem [671] 11. Field of definition of a representation [674] Appendix 1 The Transcendence of e and п [681] Appendix 2 Some Set Theory [688] Index [707]
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Part One Groups, Rings, and Modules Chapter I Groups [3] --
I Monoids [3] --
2. Groups [7] --
3. Cyclic groups [11] --
4. Normal subgroups [12] --
5. Operations of a group on a set [20] --
6. Sylow subgroups [24] --
7. Categories and functors [26] --
8. Free groups [34] --
9. Direct sums and free abelian groups [41] --
10. Finitely generated abelian groups [47] --
11. The dual group [51] --
Chapter II Rings [60] --
1. Rings and homomorphisms [60] --
2. Commutative rings [66] --
3. Localization [71] --
4. Principal rings [74] --
5. Spec of a ring [77] --
Chapter III Modules [81] --
1. Basic definitions [81] --
2. The group of homomorphisms [84] --
3. Direct products and sums of modules [86] --
4. Free modules [92] --
5. Vector spaces [93] --
6. The dual space [96] --
7. The snake lemma [100] --
8. Projective and injective modules [101] --
9. Direct and inverse limits [106] --
Chapter IV Homology [116] --
1. Complexes [117] --
2. Homology sequence [121] --
3. Euler characteristic [123] --
4. Special complexes [133] --
5. Homotopies of morphisms of complexes [137] --
6. Derived functors [141] --
7. Delta-functors [148] --
8. Bifunctors [155] --
9. Spectral sequences [163] --
Chapter V Polynomials [176] --
1. Free algebras [176] --
2. Definition of polynomials [180] --
3. Elementary properties of polynomials [185] --
4. The Euclidean algorithm [190] --
5. Partial fractions [194] --
6. Unique factorization in several variables [197] --
7. Criteria for irreducibility [200] --
8. The derivative and multiple roots [202] --
9. Symmetric polynomials [204] --
10. The resultant [206] --
11. Power series [211] --
Chapter VI Noetherian Rings and Modules [222] --
1. Basic criteria [222] --
2. Hilbert’s theorem [226] --
3. Power series are Noetherian [227] --
4. Associated primes [228] --
5. Primary decomposition [233] --
6. Nakayama’s lemma [236] --
7. Filtered and graded modules [238] --
8. The Hilbert polynomial [243] --
9. Indecomposable modules [246] --
10. Finite free resolutions [250] --
Part Two Field Theory --
Chapter VII Algebraic Extensions [265] --
1. Finite and algebraic extensions [265] --
2. Algebraic closure [271] --
3. Splitting fields and normal extensions [278] --
4. Separable extensions [281] --
5. Finite fields [287] --
6. Primitive elements [290] --
7. Purely inseparable extensions [291] --
Chapter VIII Galois Theory [300] --
1. Galois extensions [300] --
2. Examples and applications [308] --
3. Roots of unity [313] --
4. Linear independence of characters [318] --
5. The norm and trace [320] --
6. Cyclic extensions [323] --
7. Solvable and radical extensions [326] --
8. Abelian Kummer theory [328] --
9. The equation Xn- a = 0 [331] --
10. Galois cohomology [334] --
11. Non-abelian Kummer extensions [336] --
12. Algebraic independence of homomorphisms [340] --
13. The normal basis theorem [344] --
Chapter IX Extensions of Rings [355] --
1. Integral ring extensions [355] --
2. Integral Galois extensions [362] --
3. Extension of homomorphisms [368] --
Chapter X Transcendental Extensions [372] --
1. Transcendence bases [372] --
2. Hilbert’s Nullstellensatz [374] --
3. Algebraic sets [376] --
4. Noether normalization theorem [378] --
5. Linearly disjoint extensions [379] --
6. Separable extensions [382] --
7. Derivations [385] --
Chapter XI Real Fields [390] --
1. Ordered fields [390] --
2. Real fields [392] --
3. Real zeros and homomorphisms [398] --
Chapter XII Absolute Values [404] --
1. Definitions, dependence, and independence [404] --
2. Completions [407] --
3. Finite extensions [414] --
4. Valuations [417] --
5. Completions and valuations [425] --
6. Discrete valuations [426] --
7. Zeros of polynomials in complete fields [430] --
Part Three --
Linear Algebra and Representations --
Chapter XIII --
Matrices and Linear Maps [441] --
1. Matrices [441] --
2. The rank of a matrix [444] --
3. Matrices and linear maps [445] --
4. Determinants [449] --
5. Duality [458] --
6. Matrices and bilinear forms [463] --
7. Sesquilinear duality [467] --
8. The simplicity of SL2(F)/ ± 1 [472] --
9. The group SLn(F), n >=3 [476] --
10. Fitting ideals [480] --
11. Unimodular polynomial vectors [488] --
Chapter XIV Structure of Bilinear Forms [498] --
1. Preliminaries, orthogonal sums [498] --
2. Quadratic maps [501] --
3. Symmetric forms, orthogonal bases [502] --
4. Hyperbolic spaces [503] --
5. Witt’s theorem [505] --
6. The Witt group [508] --
7. Symmetric forms over ordered fields [509] --
8. The Clifford algebra [511] --
9. Alternating forms [515] --
10. The Pfaffian [517] --
11. Hermitian forms [519] --
12. The spectral theorem (hermitian case) [521] --
13. The spectral theorem (symmetric case) [524] --
Chapter XV Representation of One Endomorphism [529] --
1. Representations [529] --
2. Modules over principal rings [532] --
3. Decomposition over one endomorphism [541] --
4. The characteristic polynomial [545] --
Chapter XVI Multilinear Products [554] --
1. Tensor product [554] --
2. Basic properties [560] --
3. Flat modules [565] --
4. Extension of the base [575] --
5. Some functorial isomorphisms --
6. Tensor product of algebras [581] --
7. The tensor algebra of a module [583] --
8. Symmetric products [586] --
9. Alternating products [588] --
10. The Koszul complex [593] --
11. The Grothendieck ring [605] --
12. Universal derivations [610] --
Chapter XVII Semisimplicity [621] --
1. Matrices and linear maps over non-commutative rings [621] --
2. Conditions defining semisimplicity [625] --
3. The density theorem [626] --
4. Semisimple rings [629] --
5. Simple rings [632] --
6. Balanced modules [636] --
Chapter XVIII Representations of Finite Groups [639] --
1. Semisimplicity of the group algebra [639] --
2. Characters [641] --
3. 1-dimensional representations [645] --
4. The space of class functions [647] --
5. Orthogonality relations [651] --
6. Induced characters [659] --
7. Induced representations [661] --
8. Positive decomposition of the regular character [666] --
9. Super solvable groups [668] --
10. Brauer’s theorem [671] --
11. Field of definition of a representation [674] --
Appendix 1 The Transcendence of e and п [681] --
Appendix 2 Some Set Theory [688] --
Index [707] --
MR, 86j:00003
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