Algebra:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2005
|
| Edition: | Rev. 3. ed., [Nachdr.] |
| Series: | Graduate texts in mathematics
211 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 895 - 901 |
| Physical Description: | XV, 914 S. graph. Darst. |
| ISBN: | 038795385X 9780387953854 |
Staff View
MARC
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| 245 | 1 | 0 | |a Algebra |c Serge Lang |
| 250 | |a Rev. 3. ed., [Nachdr.] | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2005 | |
| 300 | |a XV, 914 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804136714110763008 |
|---|---|
| adam_text | CONTENTS
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
83
117
xi
XII CONTENTS
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.
Hubert 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 Two 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
26
1
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 Xn
-
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
CONTENTS
Xiií
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.
Hubert 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 Hubert 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
XIII
Matrices and Linear Maps
503
1.
Matrices
503
2.
The rank of a matrix
506
xiv
CONTENTS
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)/±
536
9.
The group SLn(F),
η
> 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.
The tensor algebra of a module
632
8.
Symmetric products
635
Chapter
XVII
Semisimplicity
641
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
CONTENTS
XV
Chapter
XVIII
Representations of Finite Groups
663
1.
Representations and semisimplicity
663
2.
Characters
667
3. 1
-dimensional representations
671
4.
The space of class functions
673
5.
Orthogonality relations
677
6.
Induced characters
686
7.
Induced representations
688
8.
Positive decomposition of the regular character
699
9.
Supersolvable groups
702
10.
Brauer s theorem
704
11.
Field of definition of a representation
710
12.
Example: GL2 over a finite field
712
Chapter
XIX
The Alternating Product
731
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
π
867
Appendix
2
Some Set Theory
875
Bibliography
895
Index
903
|
| adam_txt |
CONTENTS
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
83
117
xi
XII CONTENTS
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.
Hubert'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 Two 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
26
1
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 Xn
-
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
CONTENTS
Xiií
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.
Hubert'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 Hubert 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
XIII
Matrices and Linear Maps
503
1.
Matrices
503
2.
The rank of a matrix
506
xiv
CONTENTS
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)/±\
536
9.
The group SLn(F),
η
> 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.
The tensor algebra of a module
632
8.
Symmetric products
635
Chapter
XVII
Semisimplicity
641
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
CONTENTS
XV
Chapter
XVIII
Representations of Finite Groups
663
1.
Representations and semisimplicity
663
2.
Characters
667
3. 1
-dimensional representations
671
4.
The space of class functions
673
5.
Orthogonality relations
677
6.
Induced characters
686
7.
Induced representations
688
8.
Positive decomposition of the regular character
699
9.
Supersolvable groups
702
10.
Brauer's theorem
704
11.
Field of definition of a representation
710
12.
Example: GL2 over a finite field
712
Chapter
XIX
The Alternating Product
731
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
π
867
Appendix
2
Some Set Theory
875
Bibliography
895
Index
903 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Lang, Serge 1927-2005 |
| author_GND | (DE-588)119305119 |
| author_facet | Lang, Serge 1927-2005 |
| author_role | aut |
| author_sort | Lang, Serge 1927-2005 |
| author_variant | s l sl |
| building | Verbundindex |
| bvnumber | BV022577547 |
| callnumber-first | Q - Science |
| callnumber-label | QA154 |
| callnumber-raw | QA154.3 |
| callnumber-search | QA154.3 |
| callnumber-sort | QA 3154.3 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 200 |
| ctrlnum | (OCoLC)310466378 (DE-599)BVBBV022577547 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | Rev. 3. ed., [Nachdr.] |
| format | Book |
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| genre | 1\p (DE-588)4143389-0 Aufgabensammlung gnd-content 2\p (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Aufgabensammlung Lehrbuch |
| id | DE-604.BV022577547 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:15:34Z |
| indexdate | 2024-07-09T21:00:47Z |
| institution | BVB |
| isbn | 038795385X 9780387953854 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015783786 |
| oclc_num | 310466378 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-20 DE-703 DE-384 DE-355 DE-BY-UBR DE-634 DE-739 |
| owner_facet | DE-19 DE-BY-UBM DE-20 DE-703 DE-384 DE-355 DE-BY-UBR DE-634 DE-739 |
| physical | XV, 914 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Lang, Serge 1927-2005 Verfasser (DE-588)119305119 aut Algebra Serge Lang Rev. 3. ed., [Nachdr.] New York [u.a.] Springer 2005 XV, 914 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 211 Literaturverz. S. 895 - 901 Algebra Algebra (DE-588)4001156-2 gnd rswk-swf 1\p (DE-588)4143389-0 Aufgabensammlung gnd-content 2\p (DE-588)4123623-3 Lehrbuch gnd-content Algebra (DE-588)4001156-2 s DE-604 Graduate texts in mathematics 211 (DE-604)BV000000067 211 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015783786&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lang, Serge 1927-2005 Algebra Graduate texts in mathematics Algebra Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4001156-2 (DE-588)4143389-0 (DE-588)4123623-3 |
| title | Algebra |
| title_auth | Algebra |
| title_exact_search | Algebra |
| title_exact_search_txtP | Algebra |
| title_full | Algebra Serge Lang |
| title_fullStr | Algebra Serge Lang |
| title_full_unstemmed | Algebra Serge Lang |
| title_short | Algebra |
| title_sort | algebra |
| topic | Algebra Algebra (DE-588)4001156-2 gnd |
| topic_facet | Algebra Aufgabensammlung Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015783786&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT langserge algebra |