Normal view

## Algebra / Serge Lang.

Editor: Reading, Mass. : Addison-Wesley, c1965Descripción: xvii, 508 p. ; 24 cmTema(s): AlgebraOtra clasificación: 00A05
Contenidos:
```Part One
Groups, Rings, and Modules
Chapter I
Groups
1. Monoids [5]
2. Groups [9]
3. Cyclic groups [13]
4. Normal subgroups [14]
5. Operation of a group on a set [19]
6. Sylow subgroups [23]
7. Categories and functors [25]
8. Free groups [33]
9. Direct sums and free abelian groups [40]
10. Finitely generated abelian groups [45]
11. The dual group [50]
Chapter II
Rings
1. Rings and homomorphisms [56]
2. Commutative rings [62]
3. Localization [66]
4. Principal rings [70]
Chapter III
Modules
1. Basic definitions [74]
2. The group of homomorphisms [76]
3. Direct products and sums of modules [79]
4. Free modules [84]
5. Vector spaces [85]
6. The dual space [88]
Chapter IV
Homology
1. Complexes [94]
2. Homology sequence [95]
3. Euler characteristic [98]
4. The Jordan-Hdlder theorem [192]
Chapter V
Polynomials
1. Free algebras [106]
2. Definition of polynomials [110]
3. Elementary properties of polynomials [115]
4. The Euclidean algorithm [120]
5. Partial fractions [123]
6. Unique factorization in several variables [126]
7. Criteria for irreducibility [128]
8. The derivative and multiple roots [130]
9. Symmetric polynomials [132]
10. The resultant [135]
Chapter VI
Noetherian Rings and Modules
1. Basic criteria [142]
2. Hilbert’s theorem [144]
3. Power series [146]
4. Associated primes [148]
5. Primary decomposition [152]
Part Two
Field Theory
Chapter VII
Algebraic Extensions
1. Finite and algebraic extensions [161]
2. Algebraic closure [166]
3. Splitting fields and normal extensions [173]
4. Separable extensions [176]
5. Finite fields [182]
6. Primitive elements [185]
7. Purely inseparable extensions [186]
Chapter VIII
Galois Theory
1. Galois extensions [192]
2. Examples and applications [199]
3. Roots of unity [203]
4. Linear independence of characters [208]
5. The norm and trace [210]
6. Cyclic extensions [213]
7. Solvable and radical extensions [216]
8. Kummer theory [218]
9. The equation Xn — a = 0 [221]
10. Galois cohomology [224]
11. Algebraic independence of homomorphisms [225]
12. The normal basis theorem [229]
Chapter IX
Extensions of Rings
1. Integral ring extensions [237]
2. Integral Galois extensions [244]
3. Extension of homomorphisms [249]
Chapter X
Transcendental Extensions
1. Transcendence bases [253]
2. Hilbert’s Nulls tellensatz [255]
3. Algebraic sets [257]
4. Noether normalization theorem [260]
5. Linearly disjoint extensions [261]
6. Separable extensions [264]
7. Derivations [266]
Chapter XI
Real Fields
1. Ordered fields [271]
2. Real fields [273]
3. Real zeros and homomorphisms [278]
Chapter XII
Absolute Values
1. Definition, dependence, and independence [283]
2. Completions [286]
3. Finite extensions [292]
4. Valuations [296]
5. Completions and valuations [304]
6. Discrete valuations [305]
7. Zeros of polynomials over complete fields [308]
Part Three
Linear Algebra and Representations
Chapter XIII
Matrices and Linear Maps
1. Matrices [321]
2. The rank of a matrix [323]
3. Matrices and linear maps [324]
4. Determinants [328]
5. Duality [337]
6. Matrices and bilinear forms [342]
7. Sesquilinear duality [346]
Chapter XIV
Structure of Bilinear Forms
1. Preliminaries, orthogonal sums [354]
3. Symmetric forms, orthogonal bases [358]
4. Hyperbolic spaces [359]
5. Witt’s theorem [360]
6. The Witt group [363]
7. Symmetric forms over ordered fields [365]
8. The Clifford algebra [367]
9. Alternating forms [370]
10. The Pfaffian [372]
11. Hermitian forms [374]
12. The spectral theorem (hermitian case) [376]
13. The spectral theorem (symmetric case) [378]
Chapter XV
Representation of One Endomorphism
1. Representations [384]
2. Modules over principal rings [386]
3. Decomposition over one endomorphism [395]
4. The characteristic polynomial [399]
Chapter XVI
Multilinear Products
1. Tensor product [408]
2. Basic properties [412]
3. Extension of the base [418]
4. Tensor product of algebras [420]
5. The tensor algebra of a module [421]
6. Alternating products [424]
7. Symmetric products [427]
8. The Euler-Grothendieck ring [429]
9. Some functorial isomorphisms [431]
Chapter XVII
Semisimplicity
1. Matrices and linear maps over non-commutative rings [438]
2. Conditions defining semisimplicity [441]
3. The density theorem [443]
4. Semisimple rings [446]
5. Simple rings [448]
Chapter XVIII
Representations of Finite Groups
1. Semisimplicity of the group algebra [453]
2. Characters [455]
3. One-dimensional representations [459]
4. The space of class functions [461]
5. Orthogonality relations [465]
6. Induced characters [468]
7. Induced representations [471]
8. Positive decomposition of the regular character [475]
9. Supersolvable groups [478]
10. Brauer’s theorem [480]
11. Field of definition of a representation [485]
Appendix. The transcendence of e and п [493]
Index [501]```
Item type Home library Shelving location Call number Materials specified Status Date due Barcode Course reserves
Libros
Libros ordenados por tema 00A05A L269 (Browse shelf) Available A-2364

Bibliografía: p. xii.

Part One --
Groups, Rings, and Modules --
Chapter I --
Groups --
1. Monoids [5] --
2. Groups [9] --
3. Cyclic groups [13] --
4. Normal subgroups [14] --
5. Operation of a group on a set [19] --
6. Sylow subgroups [23] --
7. Categories and functors [25] --
8. Free groups [33] --
9. Direct sums and free abelian groups [40] --
10. Finitely generated abelian groups [45] --
11. The dual group [50] --
Chapter II --
Rings --
1. Rings and homomorphisms [56] --
2. Commutative rings [62] --
3. Localization [66] --
4. Principal rings [70] --
Chapter III --
Modules --
1. Basic definitions [74] --
2. The group of homomorphisms [76] --
3. Direct products and sums of modules [79] --
4. Free modules [84] --
5. Vector spaces [85] --
6. The dual space [88] --
Chapter IV --
Homology --
1. Complexes [94] --
2. Homology sequence [95] --
3. Euler characteristic [98] --
4. The Jordan-Hdlder theorem [192] --
Chapter V --
Polynomials --
1. Free algebras [106] --
2. Definition of polynomials [110] --
3. Elementary properties of polynomials [115] --
4. The Euclidean algorithm [120] --
5. Partial fractions [123] --
6. Unique factorization in several variables [126] --
7. Criteria for irreducibility [128] --
8. The derivative and multiple roots [130] --
9. Symmetric polynomials [132] --
10. The resultant [135] --
Chapter VI --
Noetherian Rings and Modules --
1. Basic criteria [142] --
2. Hilbert’s theorem [144] --
3. Power series [146] --
4. Associated primes [148] --
5. Primary decomposition [152] --
Part Two --
Field Theory --
Chapter VII --
Algebraic Extensions --
1. Finite and algebraic extensions [161] --
2. Algebraic closure [166] --
3. Splitting fields and normal extensions [173] --
4. Separable extensions [176] --
5. Finite fields [182] --
6. Primitive elements [185] --
7. Purely inseparable extensions [186] --
Chapter VIII --
Galois Theory --
1. Galois extensions [192] --
2. Examples and applications [199] --
3. Roots of unity [203] --
4. Linear independence of characters [208] --
5. The norm and trace [210] --
6. Cyclic extensions [213] --
7. Solvable and radical extensions [216] --
8. Kummer theory [218] --
9. The equation Xn — a = 0 [221] --
10. Galois cohomology [224] --
11. Algebraic independence of homomorphisms [225] --
12. The normal basis theorem [229] --
Chapter IX --
Extensions of Rings --
1. Integral ring extensions [237] --
2. Integral Galois extensions [244] --
3. Extension of homomorphisms [249] --
Chapter X --
Transcendental Extensions --
1. Transcendence bases [253] --
2. Hilbert’s Nulls tellensatz [255] --
3. Algebraic sets [257] --
4. Noether normalization theorem [260] --
5. Linearly disjoint extensions [261] --
6. Separable extensions [264] --
7. Derivations [266] --
Chapter XI --
Real Fields --
1. Ordered fields [271] --
2. Real fields [273] --
3. Real zeros and homomorphisms [278] --
Chapter XII --
Absolute Values --
1. Definition, dependence, and independence [283] --
2. Completions [286] --
3. Finite extensions [292] --
4. Valuations [296] --
5. Completions and valuations [304] --
6. Discrete valuations [305] --
7. Zeros of polynomials over complete fields [308] --
Part Three --
Linear Algebra and Representations --
Chapter XIII --
Matrices and Linear Maps --
1. Matrices [321] --
2. The rank of a matrix [323] --
3. Matrices and linear maps [324] --
4. Determinants [328] --
5. Duality [337] --
6. Matrices and bilinear forms [342] --
7. Sesquilinear duality [346] --
Chapter XIV --
Structure of Bilinear Forms --
1. Preliminaries, orthogonal sums [354] --
3. Symmetric forms, orthogonal bases [358] --
4. Hyperbolic spaces [359] --
5. Witt’s theorem [360] --
6. The Witt group [363] --
7. Symmetric forms over ordered fields [365] --
8. The Clifford algebra [367] --
9. Alternating forms [370] --
10. The Pfaffian [372] --
11. Hermitian forms [374] --
12. The spectral theorem (hermitian case) [376] --
13. The spectral theorem (symmetric case) [378] --
Chapter XV --
Representation of One Endomorphism --
1. Representations [384] --
2. Modules over principal rings [386] --
3. Decomposition over one endomorphism [395] --
4. The characteristic polynomial [399] --
Chapter XVI --
Multilinear Products --
1. Tensor product [408] --
2. Basic properties [412] --
3. Extension of the base [418] --
4. Tensor product of algebras [420] --
5. The tensor algebra of a module [421] --
6. Alternating products [424] --
7. Symmetric products [427] --
8. The Euler-Grothendieck ring [429] --
9. Some functorial isomorphisms [431] --
Chapter XVII --
Semisimplicity --
1. Matrices and linear maps over non-commutative rings [438] --
2. Conditions defining semisimplicity [441] --
3. The density theorem [443] --
4. Semisimple rings [446] --
5. Simple rings [448] --
Chapter XVIII --
Representations of Finite Groups --
1. Semisimplicity of the group algebra [453] --
2. Characters [455] --
3. One-dimensional representations [459] --
4. The space of class functions [461] --
5. Orthogonality relations [465] --
6. Induced characters [468] --
7. Induced representations [471] --
8. Positive decomposition of the regular character [475] --
9. Supersolvable groups [478] --
10. Brauer’s theorem [480] --
11. Field of definition of a representation [485] --
Appendix. The transcendence of e and п [493] --
Index [501] --

MR, 33 #5416

There are no comments on this title.