Basic algebra: along with a companion volume "Advanced algebra"
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2006
|
| Schriftenreihe: | Cornerstones
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXII, 717 S. graph. Darst. |
| ISBN: | 0817632484 9780817632489 |
Internformat
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| 245 | 1 | 0 | |a Basic algebra |b along with a companion volume "Advanced algebra" |c Anthony W. Knapp |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2006 | |
| 300 | |a XXII, 717 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cornerstones | |
| 650 | 4 | |a Algebra | |
| 650 | 0 | 7 | |a Algebra |0 (DE-588)4001156-2 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1804136227002122240 |
|---|---|
| adam_text | CONTENTS
Contents
of
Advanced Algebra
x
List of Figures
xi
Preface
xiii
Dependence Among Chapters
xvii
Standard Notation
xviii
Guide for the Reader
xix
I. PRELIMINARIES ABOUT THE INTEGERS,
POLYNOMIALS, AND MATRICES
1
1.
Division and Euclidean Algorithms
1
2.
Unique Factorization of Integers
4
3.
Unique Factorization of Polynomials
9
4.
Permutations and Their Signs
15
5.
Row Reduction
19
6.
Matrix Operations
24
7.
Problems
30
II. VECTOR SPACES OVER Q, M, AND
С
33
1.
Spanning, Linear Independence, and Bases
33
2.
Vector Spaces Defined by Matrices
38
3.
Linear Maps
42
4.
Dual Spaces
50
5.
Quotients of Vector Spaces
54
6.
Direct Sums and Direct Products of Vector Spaces
58
7.
Determinants
65
8.
Eigenvectors and Characteristic Polynomials
73
9.
Bases in the Infinite-Dimensional Case
77
10.
Problems
82
III. INNER-PRODUCT SPACES
88
1.
Inner Products and
Orthonormal
Sets
88
2.
Adjoints
98
3.
Spectral Theorem
104
4.
Problems 111
Contents
X.
MODULES
OVER
NONCOMMUTATIVE RINGS 544
1. Simple and Semisimple Modules 544
2.
Composition Series
551
3. Chain
Conditions
556
4.
Hom
and End for Modules
558
5.
Tensor Product for Modules
565
6.
Exact Sequences
574
7.
Problems
579
APPENDIX
583
Al.
Sets and Functions
583
A2. Equivalence Relations
589
A3.
Real Numbers
591
A4.
Complex Numbers
594
A5. Partial
Orderings
and
Zorn s
Lemma
595
A6. Cardinality
599
Hints for Solutions of Problems
603
Selected References
697
Index of Notation
699
Index
703
CONTENTS OF ADVANCED ALGEBRA
I. Transition to Modern Number Theory
II. Wedderburn-
Artin
Ring Theory
III.
Brauer
Group
IV. Homological Algebra
V. Three Theorems in Algebraic Number Theory
VI. Reinterpretation with Adeles and
Ideles
VII.
Infinite Field Extensions
VIII.
Background for Algebraic Geometry
IX. The Number Theory of Algebraic Curves
X. Methods of Algebraic Geometry
LIST OF FIGURES
2.1.
The vector space of lines
υ
+
U
in
К2
parallel to a given line
U
through the origin
55
2.2.
Factorization of linear maps via a quotient of vector spaces
56
2.3.
Three
1
-dimensional vector subspaces of M.2 such that each pair
has intersection
0 62
2.4.
Universal mapping property of a direct product of vector spaces
64
2.5.
Universal mapping property of a direct sum of vector spaces
65
3.1.
Geometric interpretation of the parallelogram law
91
3.2.
Resolution of a vector into a parallel component and an
orthogonal component
93
4.1.
Factorization of homomorphisms of groups via the quotient of a
group by a normal subgroup
132
4.2.
Universal mapping property of an external direct product of groups
136
4.3.
Universal mapping property of a direct product of groups
136
4.4.
Universal mapping property of an external direct sum of abelian
groups
138
4.5.
Universal mapping property of a direct sum of abelian groups
139
4.6.
Factorization of homomorphisms of rings via the quotient of a ring
by an ideal
146
4.7.
Substitution homomorphism for polynomials in one indeterminate
150
4.8.
Substitution homomorphism for polynomials in
η
indeterminates
156
4.9.
A square diagram
193
4.10.
Diagrams obtained by applying a covariant functor and a
contravariant
functor
193
4.11.
Universal mapping property of a product in a category
195
4.12.
Universal mapping property of a coproduct in a category
197
5.1.
Example of
a
nilpotent
matrix in Jordan form
231
5.2.
Powers of the
nilpotent
matrix in Figure
5.1 232
6.1.
Universal mapping property of a tensor product
261
6.2.
Diagrams for uniqueness of a tensor product
261
List of Figures
6.3.
Commutative diagram of a natural transformation
{
Tx
} 265
6.4.
Commutative diagram of a triple tensor product
274
6.5.
University mapping property of a tensor algebra
279
7.1.
Universal mapping property of a free group
305
7.2.
Universal mapping property of a free product
320
7.3.
An intertwining operator for two representations
330
7.4.
Equivalent group extensions
349
8.1.
Universal mapping property of the integral group ring of
G
371
8.2.
Universal mapping property of a free left
R
module
374
8.3.
Factorization of
R
homomorphisms via a quotient of
R
modules
376
8.4.
Universal mapping property of the group algebra RG
378
8.5.
Universal mapping property of the field of fractions of
R
380
8.6.
Real points of the curve y2
=
(x
-
l)x(x
+ 1) 409
8.7.
Universal mapping property of the localization of
R
at
S
428
9.1.
Closure of positive
constructible x
coordinates under
multiplication and division
465
9.2.
Closure of positive
constructible x
coordinates under square roots
466
9.3.
Construction of a regular pentagon
496
9.4.
Construction of a regular 17-gon
500
10.1.
Universal mapping property of a tensor product of a right
R
module
and a left
R
module
566
|
| adam_txt |
CONTENTS
Contents
of
Advanced Algebra
x
List of Figures
xi
Preface
xiii
Dependence Among Chapters
xvii
Standard Notation
xviii
Guide for the Reader
xix
I. PRELIMINARIES ABOUT THE INTEGERS,
POLYNOMIALS, AND MATRICES
1
1.
Division and Euclidean Algorithms
1
2.
Unique Factorization of Integers
4
3.
Unique Factorization of Polynomials
9
4.
Permutations and Their Signs
15
5.
Row Reduction
19
6.
Matrix Operations
24
7.
Problems
30
II. VECTOR SPACES OVER Q, M, AND
С
33
1.
Spanning, Linear Independence, and Bases
33
2.
Vector Spaces Defined by Matrices
38
3.
Linear Maps
42
4.
Dual Spaces
50
5.
Quotients of Vector Spaces
54
6.
Direct Sums and Direct Products of Vector Spaces
58
7.
Determinants
65
8.
Eigenvectors and Characteristic Polynomials
73
9.
Bases in the Infinite-Dimensional Case
77
10.
Problems
82
III. INNER-PRODUCT SPACES
88
1.
Inner Products and
Orthonormal
Sets
88
2.
Adjoints
98
3.
Spectral Theorem
104
4.
Problems 111
Contents
X.
MODULES
OVER
NONCOMMUTATIVE RINGS 544
1. Simple and Semisimple Modules 544
2.
Composition Series
551
3. Chain
Conditions
556
4.
Hom
and End for Modules
558
5.
Tensor Product for Modules
565
6.
Exact Sequences
574
7.
Problems
579
APPENDIX
583
Al.
Sets and Functions
583
A2. Equivalence Relations
589
A3.
Real Numbers
591
A4.
Complex Numbers
594
A5. Partial
Orderings
and
Zorn 's
Lemma
595
A6. Cardinality
599
Hints for Solutions of Problems
603
Selected References
697
Index of Notation
699
Index
703
CONTENTS OF ADVANCED ALGEBRA
I. Transition to Modern Number Theory
II. Wedderburn-
Artin
Ring Theory
III.
Brauer
Group
IV. Homological Algebra
V. Three Theorems in Algebraic Number Theory
VI. Reinterpretation with Adeles and
Ideles
VII.
Infinite Field Extensions
VIII.
Background for Algebraic Geometry
IX. The Number Theory of Algebraic Curves
X. Methods of Algebraic Geometry
LIST OF FIGURES
2.1.
The vector space of lines
υ
+
U
in
К2
parallel to a given line
U
through the origin
55
2.2.
Factorization of linear maps via a quotient of vector spaces
56
2.3.
Three
1
-dimensional vector subspaces of M.2 such that each pair
has intersection
0 62
2.4.
Universal mapping property of a direct product of vector spaces
64
2.5.
Universal mapping property of a direct sum of vector spaces
65
3.1.
Geometric interpretation of the parallelogram law
91
3.2.
Resolution of a vector into a parallel component and an
orthogonal component
93
4.1.
Factorization of homomorphisms of groups via the quotient of a
group by a normal subgroup
132
4.2.
Universal mapping property of an external direct product of groups
136
4.3.
Universal mapping property of a direct product of groups
136
4.4.
Universal mapping property of an external direct sum of abelian
groups
138
4.5.
Universal mapping property of a direct sum of abelian groups
139
4.6.
Factorization of homomorphisms of rings via the quotient of a ring
by an ideal
146
4.7.
Substitution homomorphism for polynomials in one indeterminate
150
4.8.
Substitution homomorphism for polynomials in
η
indeterminates
156
4.9.
A square diagram
193
4.10.
Diagrams obtained by applying a covariant functor and a
contravariant
functor
193
4.11.
Universal mapping property of a product in a category
195
4.12.
Universal mapping property of a coproduct in a category
197
5.1.
Example of
a
nilpotent
matrix in Jordan form
231
5.2.
Powers of the
nilpotent
matrix in Figure
5.1 232
6.1.
Universal mapping property of a tensor product
261
6.2.
Diagrams for uniqueness of a tensor product
261
List of Figures
6.3.
Commutative diagram of a natural transformation
{
Tx
} 265
6.4.
Commutative diagram of a triple tensor product
274
6.5.
University mapping property of a tensor algebra
279
7.1.
Universal mapping property of a free group
305
7.2.
Universal mapping property of a free product
320
7.3.
An intertwining operator for two representations
330
7.4.
Equivalent group extensions
349
8.1.
Universal mapping property of the integral group ring of
G
371
8.2.
Universal mapping property of a free left
R
module
374
8.3.
Factorization of
R
homomorphisms via a quotient of
R
modules
376
8.4.
Universal mapping property of the group algebra RG
378
8.5.
Universal mapping property of the field of fractions of
R
380
8.6.
Real points of the curve y2
=
(x
-
l)x(x
+ 1) 409
8.7.
Universal mapping property of the localization of
R
at
S
428
9.1.
Closure of positive
constructible x
coordinates under
multiplication and division
465
9.2.
Closure of positive
constructible x
coordinates under square roots
466
9.3.
Construction of a regular pentagon
496
9.4.
Construction of a regular 17-gon
500
10.1.
Universal mapping property of a tensor product of a right
R
module
and a left
R
module
566 |
| any_adam_object | 1 |
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| author | Knapp, Anthony W. 1941- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-search | 512 |
| dewey-sort | 3512 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| id | DE-604.BV022236823 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:34:26Z |
| indexdate | 2024-07-09T20:53:02Z |
| institution | BVB |
| isbn | 0817632484 9780817632489 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015447820 |
| oclc_num | 255519029 |
| open_access_boolean | |
| owner | DE-824 DE-706 DE-20 DE-703 DE-384 DE-83 DE-11 DE-91G DE-BY-TUM DE-188 |
| owner_facet | DE-824 DE-706 DE-20 DE-703 DE-384 DE-83 DE-11 DE-91G DE-BY-TUM DE-188 |
| physical | XXII, 717 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Cornerstones |
| spelling | Knapp, Anthony W. 1941- Verfasser (DE-588)132959690 aut Basic algebra along with a companion volume "Advanced algebra" Anthony W. Knapp Boston [u.a.] Birkhäuser 2006 XXII, 717 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cornerstones Algebra Algebra (DE-588)4001156-2 gnd rswk-swf Algebra (DE-588)4001156-2 s DE-604 Erscheint auch als Online-Ausgabe 0-8176-4529-2 Erscheint auch als Online-Ausgabe 978-0-8176-4529-8 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015447820&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Knapp, Anthony W. 1941- Basic algebra along with a companion volume "Advanced algebra" Algebra Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4001156-2 |
| title | Basic algebra along with a companion volume "Advanced algebra" |
| title_auth | Basic algebra along with a companion volume "Advanced algebra" |
| title_exact_search | Basic algebra along with a companion volume "Advanced algebra" |
| title_exact_search_txtP | Basic algebra along with a companion volume "Advanced algebra" |
| title_full | Basic algebra along with a companion volume "Advanced algebra" Anthony W. Knapp |
| title_fullStr | Basic algebra along with a companion volume "Advanced algebra" Anthony W. Knapp |
| title_full_unstemmed | Basic algebra along with a companion volume "Advanced algebra" Anthony W. Knapp |
| title_short | Basic algebra |
| title_sort | basic algebra along with a companion volume advanced algebra |
| title_sub | along with a companion volume "Advanced algebra" |
| topic | Algebra Algebra (DE-588)4001156-2 gnd |
| topic_facet | Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015447820&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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