Advanced algebra: along with a companion volume "Basic algebra"
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2007
|
| Series: | Cornerstones
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIV, 730 S. graph. Darst. |
| ISBN: | 0817645225 9780817645229 |
Staff View
MARC
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| 245 | 1 | 0 | |a Advanced algebra |b along with a companion volume "Basic algebra" |c Anthony W. Knapp |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2007 | |
| 300 | |a XXIV, 730 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Record in the Search Index
| _version_ | 1804136271861252096 |
|---|---|
| adam_text | CONTENTS
Contents of Basic Algebra
χ
Preface
xi
List of Figures
xv
Dependence among Chapters
xvi
Guide for the Reader
xvii
Notation and Terminology
xxi
I. TRANSITION TO MODERN NUMBER THEORY
1
1.
Historical Background
1
2.
Quadratic Reciprocity
8
3.
Equivalence and Reduction of Quadratic Forms
12
4.
Composition of Forms, Class Group
24
5.
Genera
31
6.
Quadratic Number Fields and Their Units
35
7.
Relationship of Quadratic Forms to Ideals
38
8.
Primes in the Progressions An
+ 1
and An
+ 3 50
9.
Dirichlet Series and
Euler
Products
56
10.
Dirichlet s Theorem on Primes in Arithmetic Progressions
61
11.
Problems
67
II. WEDDERBURN-ARTIN RING THEORY
76
1.
Historical Motivation
77
2. Semisimple
Rings and Wedderburn s Theorem
81
3.
Rings with Chain Condition and Artin s Theorem
87
4.
Wedderburn-Artin Radical
89
5.
Wedderburn s Main Theorem
94
6.
Semisimplicity and Tensor Products
104
7.
Skolem-Noether Theorem
111
8.
Double Centralizer Theorem
114
9.
Wedderburn s Theorem about Finite Division Rings
117
10.
Frobenius s Theorem about Division Algebras over the Reals
118
11.
Problems
120
viu
Contents
III.
BRAUER
GROUP
123
1. Definition
and Examples, Relative
Brauer
Group
124
2.
Factor Sets
132
3.
Crossed Products
135
4.
Hubert s Theorem
90 145
5.
Digression on Cohomology of Groups
147
6.
Relative
Brauer
Group when the Galois Group Is Cyclic
158
7.
Problems
162
IV. HOMOLOGIC
AL
ALGEBRA
166
1.
Overview
167
2.
Complexes and Additive Functors
171
3.
Long Exact Sequences
184
4.
Projectives and Injectives
192
5.
Derived Functors
202
6.
Long Exact Sequences of Derived Functors
210
7.
Ext and Tor
223
8.
Abelian Categories
232
9.
Problems
250
V. THREE THEOREMS IN ALGEBRAIC NUMBER THEORY
262
1.
Setting
262
2.
Discriminant
266
3.
Dedekind Discriminant Theorem
274
4.
Cubic Number Fields as Examples
279
5.
Dirichlet Unit Theorem
288
6.
Finiteness of the Class Number
298
7.
Problems
307
VI.
REINTERPRETATION
WITH ADELES AND IDELES
ЗІЗ
1.
p-adic Numbers
314
2.
Discrete Valuations
320
3.
Absolute Values
331
4.
Completions
342
5.
Hensel s Lemma
349
6.
Ramification Indices and Residue Class Degrees
353
7.
Special Features of Galois Extensions
368
8.
Different and Discriminant
371
9.
Global and Local Fields
382
10.
Adeles and
Ideles
388
11.
Problems
397
Contents ix
VII.
INFINITE
FIELD
EXTENSIONS
403
1. Nullstellensatz 404
2.
Transcendence Degree
408
3.
Separable and Purely Inseparable Extensions
414
4.
Krull Dimension
423
5.
Nonsingular and Singular Points
428
6.
Infinite Galois Groups
434
7.
Problems
445
VIII.
BACKGROUND FOR ALGEBRAIC GEOMETRY
447
1.
Historical Origins and Overview
448
2.
Resultant and Bezout s Theorem
451
3.
Projective
Plane Curves
456
4.
Intersection Multiplicity for a Line with a Curve
466
5.
Intersection Multiplicity for Two Curves
473
6.
General Form of Bezout s Theorem for Plane Curves
488
7. Gröbner
Bases
491
8.
Constructive Existence
499
9.
Uniqueness of Reduced
Gröbner
Bases
508
10.
Simultaneous Systems of Polynomial Equations
510
11.
Problems
516
IX. THE NUMBER THEORY OF ALGEBRAIC CURVES
520
1.
Historical Origins and Overview
520
2.
Divisors
531
3.
Genus
534
4.
Riemann-Roch Theorem
540
5.
Applications of the Riemann-Roch Theorem
552
6.
Problems
554
X. METHODS OF ALGEBRAIC GEOMETRY
558
1. Affine
Algebraic Sets and
Affine
Varieties
559
2.
Geometric Dimension
563
3.
Projective
Algebraic Sets and
Projective
Varieties
570
4.
Rational Functions and Regular Functions
579
5.
Morphisms
590
6.
Rational Maps
595
7.
Zariski s Theorem about Nonsingular Points
600
8.
Classification Questions about Irreducible Curves
604
9. Affine
Algebraic Sets for Monomial Ideals
618
10.
Hubert Polynomial in the
Affine
Case
626
x
Contents
Χ.
METHODS OF ALGEBRAIC GEOMETRY (Continued)
11.
Hubert
Polynomial in the
Projective
Case
633
12.
Intersections in
Projective
Space
635
13.
Schemes
638
14.
Problems
644
Hints for Solutions of Problems
649
Selected References
713
Index of Notation
717
Index
721
CONTENTS OF BASIC ALGEBRA
I. Preliminaries about the Integers, Polynomials, and Matrices
II
.
Vector Spaces over
Q
,
Ж
,
and
С
III. Inner-Product Spaces
IV. Groups and Group Actions
V. Theory of a Single Linear Transformation
VI. Multilinear Algebra
VII.
Advanced Group Theory
VIII.
Commutative Rings and Their Modules
IX. Fields and Galois Theory
X. Modules over
Noncommutative
Rings
|
| adam_txt |
CONTENTS
Contents of Basic Algebra
χ
Preface
xi
List of Figures
xv
Dependence among Chapters
xvi
Guide for the Reader
xvii
Notation and Terminology
xxi
I. TRANSITION TO MODERN NUMBER THEORY
1
1.
Historical Background
1
2.
Quadratic Reciprocity
8
3.
Equivalence and Reduction of Quadratic Forms
12
4.
Composition of Forms, Class Group
24
5.
Genera
31
6.
Quadratic Number Fields and Their Units
35
7.
Relationship of Quadratic Forms to Ideals
38
8.
Primes in the Progressions An
+ 1
and An
+ 3 50
9.
Dirichlet Series and
Euler
Products
56
10.
Dirichlet's Theorem on Primes in Arithmetic Progressions
61
11.
Problems
67
II. WEDDERBURN-ARTIN RING THEORY
76
1.
Historical Motivation
77
2. Semisimple
Rings and Wedderburn's Theorem
81
3.
Rings with Chain Condition and Artin's Theorem
87
4.
Wedderburn-Artin Radical
89
5.
Wedderburn's Main Theorem
94
6.
Semisimplicity and Tensor Products
104
7.
Skolem-Noether Theorem
111
8.
Double Centralizer Theorem
114
9.
Wedderburn's Theorem about Finite Division Rings
117
10.
Frobenius's Theorem about Division Algebras over the Reals
118
11.
Problems
120
viu
Contents
III.
BRAUER
GROUP
123
1. Definition
and Examples, Relative
Brauer
Group
124
2.
Factor Sets
132
3.
Crossed Products
135
4.
Hubert's Theorem
90 145
5.
Digression on Cohomology of Groups
147
6.
Relative
Brauer
Group when the Galois Group Is Cyclic
158
7.
Problems
162
IV. HOMOLOGIC
AL
ALGEBRA
166
1.
Overview
167
2.
Complexes and Additive Functors
171
3.
Long Exact Sequences
184
4.
Projectives and Injectives
192
5.
Derived Functors
202
6.
Long Exact Sequences of Derived Functors
210
7.
Ext and Tor
223
8.
Abelian Categories
232
9.
Problems
250
V. THREE THEOREMS IN ALGEBRAIC NUMBER THEORY
262
1.
Setting
262
2.
Discriminant
266
3.
Dedekind Discriminant Theorem
274
4.
Cubic Number Fields as Examples
279
5.
Dirichlet Unit Theorem
288
6.
Finiteness of the Class Number
298
7.
Problems
307
VI.
REINTERPRETATION
WITH ADELES AND IDELES
ЗІЗ
1.
p-adic Numbers
314
2.
Discrete Valuations
320
3.
Absolute Values
331
4.
Completions
342
5.
Hensel's Lemma
349
6.
Ramification Indices and Residue Class Degrees
353
7.
Special Features of Galois Extensions
368
8.
Different and Discriminant
371
9.
Global and Local Fields
382
10.
Adeles and
Ideles
388
11.
Problems
397
Contents ix
VII.
INFINITE
FIELD
EXTENSIONS
403
1. Nullstellensatz 404
2.
Transcendence Degree
408
3.
Separable and Purely Inseparable Extensions
414
4.
Krull Dimension
423
5.
Nonsingular and Singular Points
428
6.
Infinite Galois Groups
434
7.
Problems
445
VIII.
BACKGROUND FOR ALGEBRAIC GEOMETRY
447
1.
Historical Origins and Overview
448
2.
Resultant and Bezout's Theorem
451
3.
Projective
Plane Curves
456
4.
Intersection Multiplicity for a Line with a Curve
466
5.
Intersection Multiplicity for Two Curves
473
6.
General Form of Bezout's Theorem for Plane Curves
488
7. Gröbner
Bases
491
8.
Constructive Existence
499
9.
Uniqueness of Reduced
Gröbner
Bases
508
10.
Simultaneous Systems of Polynomial Equations
510
11.
Problems
516
IX. THE NUMBER THEORY OF ALGEBRAIC CURVES
520
1.
Historical Origins and Overview
520
2.
Divisors
531
3.
Genus
534
4.
Riemann-Roch Theorem
540
5.
Applications of the Riemann-Roch Theorem
552
6.
Problems
554
X. METHODS OF ALGEBRAIC GEOMETRY
558
1. Affine
Algebraic Sets and
Affine
Varieties
559
2.
Geometric Dimension
563
3.
Projective
Algebraic Sets and
Projective
Varieties
570
4.
Rational Functions and Regular Functions
579
5.
Morphisms
590
6.
Rational Maps
595
7.
Zariski 's Theorem about Nonsingular Points
600
8.
Classification Questions about Irreducible Curves
604
9. Affine
Algebraic Sets for Monomial Ideals
618
10.
Hubert Polynomial in the
Affine
Case
626
x
Contents
Χ.
METHODS OF ALGEBRAIC GEOMETRY (Continued)
11.
Hubert
Polynomial in the
Projective
Case
633
12.
Intersections in
Projective
Space
635
13.
Schemes
638
14.
Problems
644
Hints for Solutions of Problems
649
Selected References
713
Index of Notation
717
Index
721
CONTENTS OF BASIC ALGEBRA
I. Preliminaries about the Integers, Polynomials, and Matrices
II
.
Vector Spaces over
Q
,
Ж
,
and
С
III. Inner-Product Spaces
IV. Groups and Group Actions
V. Theory of a Single Linear Transformation
VI. Multilinear Algebra
VII.
Advanced Group Theory
VIII.
Commutative Rings and Their Modules
IX. Fields and Galois Theory
X. Modules over
Noncommutative
Rings |
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| id | DE-604.BV022268801 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:45:26Z |
| indexdate | 2024-07-09T20:53:45Z |
| institution | BVB |
| isbn | 0817645225 9780817645229 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015479323 |
| oclc_num | 255404979 |
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| physical | XXIV, 730 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Cornerstones |
| spelling | Knapp, Anthony W. 1941- Verfasser (DE-588)132959690 aut Advanced algebra along with a companion volume "Basic algebra" Anthony W. Knapp Boston [u.a.] Birkhäuser 2007 XXIV, 730 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cornerstones Álgebra abstrata larpcal Algebraic number theory Algebra (DE-588)4001156-2 gnd rswk-swf Algebra (DE-588)4001156-2 s DE-604 Erscheint auch als Online-Ausgabe 978-0-8176-4613-4 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015479323&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Knapp, Anthony W. 1941- Advanced algebra along with a companion volume "Basic algebra" Álgebra abstrata larpcal Algebraic number theory Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4001156-2 |
| title | Advanced algebra along with a companion volume "Basic algebra" |
| title_auth | Advanced algebra along with a companion volume "Basic algebra" |
| title_exact_search | Advanced algebra along with a companion volume "Basic algebra" |
| title_exact_search_txtP | Advanced algebra along with a companion volume "Basic algebra" |
| title_full | Advanced algebra along with a companion volume "Basic algebra" Anthony W. Knapp |
| title_fullStr | Advanced algebra along with a companion volume "Basic algebra" Anthony W. Knapp |
| title_full_unstemmed | Advanced algebra along with a companion volume "Basic algebra" Anthony W. Knapp |
| title_short | Advanced algebra |
| title_sort | advanced algebra along with a companion volume basic algebra |
| title_sub | along with a companion volume "Basic algebra" |
| topic | Álgebra abstrata larpcal Algebraic number theory Algebra (DE-588)4001156-2 gnd |
| topic_facet | Álgebra abstrata Algebraic number theory Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015479323&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT knappanthonyw advancedalgebraalongwithacompanionvolumebasicalgebra |