A concrete approach to abstract algebra: from the integers to the insolvability of the quintic
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam [u.a.]
Elsevier
2010
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XIX, 700 S. graph. Darst. |
| ISBN: | 9780123749413 |
Internformat
MARC
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| 245 | 1 | 0 | |a A concrete approach to abstract algebra |b from the integers to the insolvability of the quintic |c Jeffrey Bergen |
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2010 | |
| 300 | |a XIX, 700 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Algebra, Abstract | |
| 650 | 0 | 7 | |a Universelle Algebra |0 (DE-588)4061777-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Universelle Algebra |0 (DE-588)4061777-4 |D s |
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Datensatz im Suchindex
| _version_ | 1804142796627509248 |
|---|---|
| adam_text | Contents
Preface
...............................................................................xi
A User s Guide
......................................................................xv
Acknowledgments
...................................................................xix
Chapter
1
What This Book Is about and Who This Book Is for
.......................1
1.1
Algebra
...........................................................................2
1.1.1
Finding Roots of Polynomials
...................................................2
1.1.2
Existence of Roots of Polynomials
..............................................4
1.1.3
Solving Linear Equations
........................................................5
1.2
Geometry
........................................................................6
1.2.1
Ruler and Compass Constructions
...............................................6
1.3
Trigonometry
....................................................................7
1.3.1
Rational Values of Trigonometric Functions
....................................7
1.4
Precalculus
.......................................................................8
1.4.1
Recognizing Polynomials Using Data
...........................................8
1.5
Calculus
.........................................................................10
1.5.1
Partial Fraction Decomposition
................................................ 10
1.5.2
Detecting Multiple Roots of Polynomials
..................................... 12
Exercises for Chapter
1 ........................................................14
Chapter
2
Proof and Intuition
.......................................................19
2.1
The Well Ordering Principle
...................................................20
2.2
Proof by Contradiction
.........................................................26
2.3
Mathematical Induction
........................................................29
Mathematical Induction
—
First Version
.......................................30
Mathematical Induction
—
First Version Revisited
............................32
Mathematical Induction
—
Second Version
....................................37
Exercises for Sections
2.1, 2.2,
and
2.3.......................................37
2.4
Functions and Binary Operations
..............................................46
Exercises for Section
2.4.......................................................56
VII
Contents
Chapter
3
The Integers
.............................................................61
3.1
Prime Numbers
.................................................................61
3.2
Unique Factorization
...........................................................64
3.3
Division Algorithm
.............................................................67
Exercises for Sections
3.1, 3.2,
and
3.3.......................................71
3.4
Greatest Common Divisors
....................................................76
3.5
Euclidean Algorithm
...........................................................79
Exercises for Sections
3.4
and
3.5.............................................91
Chapter
4
The Rational Numbers and the Real Numbers
............................97
4.1
Rational Numbers
..............................................................97
4.2
Intermediate Value Theorem
................................................. 105
Exercises for Sections
4.1
and
4.2........................................... 113
4.3
Equivalence Relations
........................................................ 118
Exercises for Section
4.3..................................................... 128
Chapter
5
The Complex Numbers
................................................. 737
5.1
Complex Numbers
........................................................... 137
5.2
Fields and Commutative Rings
..............................................140
Exercises for Sections
5.1
and
5.2........................................... 148
5.3
Complex Conjugation
........................................................ 154
5.4
Automorphisms and Roots of Polynomials
................................. 163
Exercises for Sections
5.3
and
5.4........................................... 169
5.5
Groups of Automorphisms of Commutative Rings
......................... 177
Exercises for Section
5.5..................................................... 182
Chapter
6
The Fundamental Theorem of Algebra
................................. 189
6.1
Representing Real Numbers and Complex Numbers Geometrically
......189
6.2
Rectangular and Polar Form
................................................. 199
Exercises for Sections
6.1
and
6.2...........................................203
6.3
Demoivre s Theorem and Roots of Complex Numbers
....................208
6.4
A Proof of the Fundamental Theorem of Algebra
..........................215
Exercises for Sections
6.3
and
6.4...........................................222
Chapter
7
The Integers Modulo
η
.................................................227
7.1
Definitions and Basic Properties
.............................................227
7.2
Zero Divisors and Invertible Elements
......................................233
Exercises for Sections
7.1
and
7.2...........................................241
7.3
The
Euler
φ
Function
.........................................................248
7.4
Polynomials with Coefficients in
Z„ ........................................256
Exercises for Sections
7.3
and
7.4...........................................260
Vill
Contents
Chapter
8
Croup Theory
..........................................................265
8.1
Definitions and Examples
....................................................265
I. Commutative Rings and Fields under Addition
........................266
II. Invertible Elements in Commutative Rings under Multiplication
.....266
III. Bijections of Sets
.........................................................267
Exercises for Section
8.1.....................................................288
8.2
Theorems of
Lagrange
and Sylow
...........................................294
Exercises for Section
8.2.....................................................318
8.3
Solvable Groups
..............................................................322
Exercises for Section
8.3.....................................................342
8.4
Symmetric Groups
...........................................................347
Exercises for Section
8.4.....................................................361
Chapter
9
Polynomials over the Integers and Rationals
............................365
9.1
Integral Domains and Homomorphisms of Rings
..........................365
Exercises for Section
9.1.....................................................374
9.2
Rational Root Test and Irreducible Polynomials
............................379
Exercises for Section
9.2.....................................................387
9.3
Gauss Lemma and Eisenstein s Criterion
..................................390
Exercises for Section
9.3.....................................................397
9.4
Reduction Modulo
ρ
..........................................................398
Exercises for Section
9.4.....................................................408
Chapter
10
Roots of Polynomials of Degree Less than
5..........................411
10.1
Finding Roots of Polynomials of Small Degree
............................411
10.2
A Brief Look at Some Consequences of Galois Work
.....................418
Exercises for Sections
10.1
and
10.2........................................420
Chapter
11
Rational Values of Trigonometric Functions
...........................423
11.1
Values of Trigonometric Functions
..........................................424
Exercises for Section
11.1 ...................................................433
Chapter
12
Polynomials over Arbitrary Fields
.....................................437
12.1
Similarities between Polynomials and Integers
.............................437
12.2
Division Algorithm
...........................................................444
Exercises for Sections
12.1
and
12.2........................................453
12.3
Irreducible and Minimum Polynomials
.....................................457
12.4
Euclidean Algorithm and Greatest Common Divisors
......................460
Exercises for Sections
12.3
and
12.4........................................470
12.5
Formal Derivatives and Multiple Roots
.....................................474
Exercises for Section
12.5...................................................484
IX
Contents
Chapter
13
Difference Functions and Partial Fractions
............................487
13.1
Difference Functions
.........................................................488
13.2
Polynomials and Mathematical Induction
...................................499
Exercises for Sections
13.1
and
13.2........................................504
13.3
Partial Fraction Decomposition
..............................................510
Exercises for Section
13.3...................................................523
Chapter
14
An Introduction to Linear Algebra and Vector Spaces
.................527
14.1
Examples, Examples, Examples, and a Definition
..........................527
Exercises for Section
14.1 ...................................................538
14.2
Spanning Sets and Linear Independence
....................................540
14.3
Basis and Dimension
.........................................................548
Exercises for Sections
14.2
and
14.3........................................555
14.4
Subspaces and Linear Equations
............................................560
Exercises for Section
14.4...................................................568
Chapter
15
Degrees and Galois Croups of Field Extensions
.......................573
15.1
Degrees of Field Extensions
.................................................573
Exercises for Section
15.1 ...................................................590
15.2
Simple Extensions
............................................................594
15.3
Splitting Fields and Their Galois Groups
...................................599
Exercises for Sections
15.2
and
15.3........................................615
Chapter
16
Geometric Constructions
.............................................623
16.1
Constructible
Points and
Constructible
Real Numbers
.....................623
16.2
The Impossibility of Trisecting Angles
......................................639
Exercises for Sections
16.1
and
16.2........................................643
Chapter
17
/»solvability of the Quintic
............................................645
17.1
Radical Extensions and Their Galois Groups
...............................645
17.2
A Proof of the Insolvability of the Quintic
..................................657
Exercises for Sections
17.1
and
17.2........................................660
17.3
Kronecker s Theorem
........................................................663
Exercises for Section
17.3...................................................678
Bibliography
.......................................................................685
Index
..............................................................................687
A CONCRETE APPROACH TO
ABSTRACT ALGEBRA
From the Integers to the Insolvability of the Quintic
Beginning with a concrete and thorough examination of familiar objects
like integers, rational numbers, real numbers, complex numbers, complex
conjugation, and polynomials, in this unique approach, the author builds upon
these familiar objects and then uses them to introduce and motivate advanced
concepts in algebra in a manner that is easier to understand for most students.
A Concrete Approach to Abstract Algebra begins with integers and introduces difficult topics
after a solid introduction presenting topics that are motivated by down to earth questions
that arise in courses in algebra, geometry, trigonometry, and calculus. It also includes more
topics of interest to teachers, and ends with a proof of the insolvability of the quintic.
•
Presents a more natural rings first approach that uses concrete examples and
concepts to motivate and guide the discussion of abstract algebra as it leads the
student to the abstract material of the course
•
Bridges the gap for students by showing how most of the concepts within an abstract
algebra course are actually tools needed to solve well-known problems
•
Builds on relatively familiar material (integers, polynomials) and moves onto
more abstract topics, while providing a historical approach of introducing groups as
automorphisms
•
Exercises provide a balanced blend of difficulty levels, while the quantity allows the
instructor a latitude of choices
Cover ¡mage:
¡Stockphoto
ACADEMIC PRESS
An imprint of
Elsevier
elsevierdirectcom
ISBN:
476-0-12-374441-3
90000
749413
|
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| spelling | Bergen, Jeffrey 1955- Verfasser (DE-588)141573694 aut A concrete approach to abstract algebra from the integers to the insolvability of the quintic Jeffrey Bergen Amsterdam [u.a.] Elsevier 2010 XIX, 700 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Algebra, Abstract Universelle Algebra (DE-588)4061777-4 gnd rswk-swf Universelle Algebra (DE-588)4061777-4 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020208531&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020208531&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Bergen, Jeffrey 1955- A concrete approach to abstract algebra from the integers to the insolvability of the quintic Algebra, Abstract Universelle Algebra (DE-588)4061777-4 gnd |
| subject_GND | (DE-588)4061777-4 |
| title | A concrete approach to abstract algebra from the integers to the insolvability of the quintic |
| title_auth | A concrete approach to abstract algebra from the integers to the insolvability of the quintic |
| title_exact_search | A concrete approach to abstract algebra from the integers to the insolvability of the quintic |
| title_full | A concrete approach to abstract algebra from the integers to the insolvability of the quintic Jeffrey Bergen |
| title_fullStr | A concrete approach to abstract algebra from the integers to the insolvability of the quintic Jeffrey Bergen |
| title_full_unstemmed | A concrete approach to abstract algebra from the integers to the insolvability of the quintic Jeffrey Bergen |
| title_short | A concrete approach to abstract algebra |
| title_sort | a concrete approach to abstract algebra from the integers to the insolvability of the quintic |
| title_sub | from the integers to the insolvability of the quintic |
| topic | Algebra, Abstract Universelle Algebra (DE-588)4061777-4 gnd |
| topic_facet | Algebra, Abstract Universelle Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020208531&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020208531&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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