Elements of algebra: geometry, numbers, equations
This book is a concise, self-contained introduction to abstract algebra that stresses its unifying role in geometry and number theory. Classical results in these fields, such as the straightedge-and-compass constructions and their relation to Fermat primes, are used to motivate and illustrate algebr...
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
1994
|
| Series: | Undergraduate texts in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | This book is a concise, self-contained introduction to abstract algebra that stresses its unifying role in geometry and number theory. Classical results in these fields, such as the straightedge-and-compass constructions and their relation to Fermat primes, are used to motivate and illustrate algebraic techniques. Classical algebra itself is used to motivate the problem of solvability by radicals and its solution via Galois theory. This historical approach has at least two advantages: On the one hand it shows that abstract concepts have concrete roots, and on the other it demonstrates the power of new concepts to solve old problems. Algebra has a pedigree stretching back at least as far as Euclid, but today its connections with other parts of mathematics are often neglected or forgotten. By developing algebra out of classical number theory and geometry and reviving these connections, the author has made this book useful to beginners and experts alike. The lively style and clear exposition make it a pleasure to read and to learn from. |
| Physical Description: | XI, 181 S. graph. Darst. |
| ISBN: | 3540942904 0387942904 |
Staff View
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| 520 | 3 | |a This book is a concise, self-contained introduction to abstract algebra that stresses its unifying role in geometry and number theory. Classical results in these fields, such as the straightedge-and-compass constructions and their relation to Fermat primes, are used to motivate and illustrate algebraic techniques. Classical algebra itself is used to motivate the problem of solvability by radicals and its solution via Galois theory. This historical approach has at least two advantages: On the one hand it shows that abstract concepts have concrete roots, and on the other it demonstrates the power of new concepts to solve old problems. Algebra has a pedigree stretching back at least as far as Euclid, but today its connections with other parts of mathematics are often neglected or forgotten. By developing algebra out of classical number theory and geometry and reviving these connections, the author has made this book useful to beginners and experts alike. The lively style and clear exposition make it a pleasure to read and to learn from. | |
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Record in the Search Index
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|---|---|
| adam_text |
CONTENTS
PREFACE
VII
CHAPTER
1.
ALGEBRA
AND
GEOMETRY
1
1.1
ALGEBRAIC
PROBLEMS
1
1.2
STRAIGHTEDGE
AND
COMPASS
CONSTRUCTIONS
1
1.3
THE
CONSTRUCTIBLE
NUMBERS
5
1.4
SOME
FAMOUS
CONSTRUCTIBLE
FIGURES
7
1.5
THE
CLASSICAL
CONSTRUCTION
PROBLEMS
10
1.6
QUADRATIC
AND
CUBIC
EQUATIONS
11
1.7
QUARTIC
EQUATIONS
13
1.8
SOLUTION
BY
RADICALS
14
1.9
DISCUSSION
15
CHAPTER
2.
THE
RATIONAL
NUMBERS
18
2.1
NATURAL
NUMBERS
18
2.2
INTEGERS
AND
RATIONAL
NUMBERS
20
2.3
DIVISIBILITY
22
2.4
THE
EUCLIDEAN
ALGORITHM
23
2.5
UNIQUE
PRIME
FACTORISATION
25
2.6
CONGRUENCES
26
2.7
RINGS
AND
FIELDS
OF
CONGRUENCE
CLASSES
28
2.8*
THE
THEOREMS
OF
FERMAT
AND
EULER
29
2.9*
FRACTIONS
AND
THE
EULER
PHI
FUNCTION
32
2.10
DISCUSSION
34
CHAPTER
3.
NUMBERS
IN
GENERAL
38
3.1
IRRATIONAL
NUMBERS
38
3.2
EXISTENCE
AND
MEANING
OF
IRRATIONAL
NUMBERS
40
3.3
THE
REAL
NUMBERS
41
3.4
ARITHMETIC
AND
RATIONAL
FUNCTIONS
ON
R
42
3.5
CONTINUITY
AND
COMPLETENESS
44
3.6
COMPLEX
NUMBERS
46
3.7
REGULAR
POLYGONS
48
3.8
THE
FUNDAMENTAL
THEOREM
OF
ALGEBRA
51
3.9
DISCUSSION
53
CHAPTER
4.
POLYNOMIALS
57
4.1
POLYNOMIALS
OVER
A
FIELD
57
4.2
DIVISIBILITY
58
4.3
UNIQUE
FACTORISATION
61
4.4
CONGRUENCES
62
X
CONTENTS
4.5
THE
FIELDS
F(A)
63
4.6
GAUSS
'
S
LEMMA
65
4.7
EISENSTEIN
'
S
IRREDUCIBILITY
CRITERION
67
4.8*
CYCLOTOMIC
POLYNOMIALS
69
4.9*
IRREDUCIBILITY
OF
CYCLOTOMIC
POLYNOMIALS
71
4.10
DISCUSSION
73
CHAPTER
5.
FIELDS
76
5.1
THE
STORY
SO
FAR
76
5.2
ALGEBRAIC
NUMBERS
AND
FIELDS
76
5.3
ALGEBRAIC
ELEMENTS
OVER
AN
ARBITRARY
FIELD
77
5.4
DEGREE
OF
A
FIELD
OVER
A
SUBFIELD
78
5.5
DEGREE
OF
AN
ITERATED
EXTENSION
81
5.6
DEGREE
OF
CONSTRUCTIBLE
NUMBERS
83
5.7*
REGULAR
N-GONS
85
5.8
DISCUSSION
86
CHAPTER
6.
ISOMORPHISMS
89
6.1
RING
AND
FIELD
ISOMORPHISMS
89
6.2
ISOMORPHISMS
OF
Q(A)
AND
F(A)
91
6.3
EXTENDING
FIELDS
AND
ISOMORPHISMS
94
6.4
AUTOMORPHISMS
AND
GROUPS
97
6.5*
FUNCTION
FIELDS
AND
SYMMETRIC
FUNCTIONS
98
6.6*
CYCLOTOMIC
FIELDS
100
6.7*
THE
CHINESE
REMAINDER
THEOREM
101
6.8
HOMOMORPHISMS
AND
QUOTIENT
RINGS
103
6.9
DISCUSSION
105
CHAPTER
7.
GROUPS
108
7.1
WHY
GROUPS?
108
7.2
CAYLEY
'
S
THEOREM
109
7.3
ABELIAN
GROUPS
111
7.4
DIHEDRAL
GROUPS
112
7.5*
PERMUTATION
GROUPS
115
7.6*
PERMUTATION
GROUPS
IN
GEOMETRY
116
7.7
SUBGROUPS
AND
COSETS
119
7.8
NORMAL
SUBGROUPS
121
7.9
HOMOMORPHISMS
122
7.10
DISCUSSION
125
CHAPTER
8.
GALOIS
THEORY
OF
UNSOLVABILITY
128
8.1
GALOIS
GROUPS
128
8.2
SOLUTION
BY
RADICALS
130
8.3
STRUCTURE
OF
RADICAL
EXTENSIONS
132
CONTENTS
XI
8.4
NONEXISTENCE
OF
SOLUTIONS
BY
RADICALS
WHEN
N
5
134
8.5*
QUINTICS
WITH
INTEGER
COEFFICIENTS
136
8.6*
UNSOLVABLE
QUINTIC
EQUATIONS
WITH
INTEGER
COEFFICIENTS
138
8.7*
PRIMITIVE
ROOTS
139
8.8
FINITE
ABELIAN
GROUPS
141
8.9
DISCUSSION
143
CHAPTER
9.
GALOIS
THEORY
OF
SOLVABILITY
146
9.1
THE
THEOREM
OF
THE
PRIMITIVE
ELEMENT
146
9.2
CONJUGATE
FIELDS
AND
SPLITTING
FIELDS
148
9.3
FIXED
FIELDS
150
9.4
CONJUGATE
INTERMEDIATE
FIELDS
152
9.5
NORMAL
EXTENSIONS
WITH
SOLVABLE
GALOIS
GROUP
154
9.6
CYCLIC
EXTENSIONS
155
9.7
CONSTRUCTION
OF
THE
RADICAL
EXTENSION
156
9.8
CONSTRUCTION
OF
REGULAR
P-GONS
157
9.9*
DIVISION
OF
ARBITRARY
ANGLES
159
9.10
DISCUSSION
160
REFERENCES
162
INDEX
170 |
| any_adam_object | 1 |
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.02 |
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| isbn | 3540942904 0387942904 |
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| spelling | Stillwell, John 1942- Verfasser (DE-588)128427264 aut Elements of algebra geometry, numbers, equations John Stillwell New York u.a. Springer 1994 XI, 181 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics This book is a concise, self-contained introduction to abstract algebra that stresses its unifying role in geometry and number theory. Classical results in these fields, such as the straightedge-and-compass constructions and their relation to Fermat primes, are used to motivate and illustrate algebraic techniques. Classical algebra itself is used to motivate the problem of solvability by radicals and its solution via Galois theory. This historical approach has at least two advantages: On the one hand it shows that abstract concepts have concrete roots, and on the other it demonstrates the power of new concepts to solve old problems. Algebra has a pedigree stretching back at least as far as Euclid, but today its connections with other parts of mathematics are often neglected or forgotten. By developing algebra out of classical number theory and geometry and reviving these connections, the author has made this book useful to beginners and experts alike. The lively style and clear exposition make it a pleasure to read and to learn from. Algèbre ram Algebra Algebra (DE-588)4001156-2 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Universelle Algebra (DE-588)4061777-4 gnd rswk-swf Algebra (DE-588)4001156-2 s DE-604 Universelle Algebra (DE-588)4061777-4 s Geometrie (DE-588)4020236-7 s DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006417413&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stillwell, John 1942- Elements of algebra geometry, numbers, equations Algèbre ram Algebra Algebra (DE-588)4001156-2 gnd Geometrie (DE-588)4020236-7 gnd Universelle Algebra (DE-588)4061777-4 gnd |
| subject_GND | (DE-588)4001156-2 (DE-588)4020236-7 (DE-588)4061777-4 |
| title | Elements of algebra geometry, numbers, equations |
| title_auth | Elements of algebra geometry, numbers, equations |
| title_exact_search | Elements of algebra geometry, numbers, equations |
| title_full | Elements of algebra geometry, numbers, equations John Stillwell |
| title_fullStr | Elements of algebra geometry, numbers, equations John Stillwell |
| title_full_unstemmed | Elements of algebra geometry, numbers, equations John Stillwell |
| title_short | Elements of algebra |
| title_sort | elements of algebra geometry numbers equations |
| title_sub | geometry, numbers, equations |
| topic | Algèbre ram Algebra Algebra (DE-588)4001156-2 gnd Geometrie (DE-588)4020236-7 gnd Universelle Algebra (DE-588)4061777-4 gnd |
| topic_facet | Algèbre Algebra Geometrie Universelle Algebra |
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