Galois theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton ; London ; New York
CRC Press, Taylor & Francis Group
[2015]
|
| Edition: | Fourth edition |
| Series: | A Chapman & Hall book
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Previous edition, 2004. - Includes bibliographical references (pages 309-314) and index |
| Physical Description: | xxii, 321 Seiten Illustrationen, Diagramme 24 cm |
| ISBN: | 9781482245820 |
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| 100 | 1 | |a Stewart, Ian |d 1945- |0 (DE-588)124091598 |4 aut | |
| 245 | 1 | 0 | |a Galois theory |c Ian Stewart, University of Warwick, Coventry, UK |
| 250 | |a Fourth edition | ||
| 264 | 1 | |a Boca Raton ; London ; New York |b CRC Press, Taylor & Francis Group |c [2015] | |
| 264 | 4 | |c © 2015 | |
| 300 | |a xxii, 321 Seiten |b Illustrationen, Diagramme |c 24 cm | ||
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Record in the Search Index
| _version_ | 1804176105694822400 |
|---|---|
| adam_text | CONTENTS
ACKNOWLEDGEMENTS
PREFACE
TO THE
FIRST EDITION
PREFACE
TO THE
SECOND EDITION
PREFACE
TO THE
THIRD EDITION
PREFACE
TO THE
FOURTH
EDITION
HISTORICAL
INTRODUCTION
%1
XLLL
XV
XVLL
XXI
1
1 CLASSICAL ALGEBRA
17
1.1
COMPLEX NUMBERS
..................
........
18
1.2
SUBFIELDS
AND
SUBRINGS
OF
THE
COMPLEX
NUMBERS
..........
18
1.3
SOLVING EQUATIONS
..........................
22
1.4
SOLUTION BY RADICALS
. .....
... . ....
...........
24
2
THE FUNDAMENTAL
THEOREM
OF
ALGEBRA
35
2.1
POLYNOMIALS
.............................
35
2.2
FUNDAMENTAL
THEOREM
OF
ALGEBRA
..................
39
2.3
IMPLICATIONS
.............................
42
3
FACTORISATION
OF
POLYNOMIALS
47
3.1 THE EUCLIDEAN
ALGORITHM
......................
47
3.2 IRREDUCIBILITY
.............................
51
3.3
GAUSS S
LEMMA
............................
54
3.4 EISENSTEIN S
CRITERION
.........................
55
3.5
REDUCTION
MODULO
P.........................
57
3.6
ZEROS
OF
POLYNOMIALS
.......................
58
4
FIELD EXTENSIONS
63
4.1 FIELD EXTENSIONS
...........
.............
..
63
4.2 RATIONAL
EXPRESSIONS
.........................
66
4.3 SIMPLE EXTENSIONS
........................
..
67
VII
$
BIIOTHEK
DEUTSCHES
MUSEUM
T?
HLUENCHE A
VIII
CONTENTS
5
SIMPLE EXTENSIONS
71
5.1 ALGEBRAIC
AND
TRANSCENDENTAL EXTENSIONS
....... .......
71
5.2
THE
MINIMAL
POLYNOMIAL
..................
.....
72
5.3
SIMPLE ALGEBRAIC EXTENSIONS
......
.... ...........
73
5.4
CLASSIFYING SIMPLE EXTENSIONS
. ... ................
75
6
THE DEGREE
OF
AN
EXTENSION
79
6.1
DEFINITION
OF
THE
DEGREE
........
.............
..
79
6.2 THE TOWER LAW
............... ......
.......
80
7
RULER-AND-COMPASS
CONSTRUCTIONS
87
7.1 APPROXIMATE
CONSTRUCTIONS
AND
MORE
GENERAL
INSTRUMENTS
.....
89
7.2 CONSTRUCTIONS IN
C...........
...............
90
7.3 SPECIFIC CONSTRUCTIONS
.
...............
.......
94
7.4 IMPOSSIBILITY
PROOFS
.........
................
99
7.5
CONSTRUCTION FROM
A
GIVEN
SET
OF
POINTS
..........
....
101
8 THE IDEA BEHIND
GALOIS THEORY
107
8.1 A
FIRST
LOOK
AT
GALOIS
THEORY
....................
108
8.2 GALOIS
GROUPS
ACCORDING
TO
GALOIS
............
.....
108
8.3
HOW
TO
USE
THE
GALOIS GROUP
.......
.............
110
8.4 THE ABSTRACT
SETTING
..........
.......... .....
111
8.5 POLYNOMIALS
AND
EXTENSIONS
...... ......
.........
112
8.6 THE
GALOIS CORRESPONDENCE
.....
. ...............
114
8.7 DIET
GALOIS
... ....... ...........
..... ....
116
8.8
NATURAL IRRATIONALITIES
......
............ .......
121
9
NORMALITY
AND
SEPARABILITY
129
9.1
SPLITTING
FIELDS
............
...........
.....
129
9.2 NORMALITY
...................
.........
132
9.3 SEPARABILITY
...... .............
.......... .
133
10
COUNTING PRINCIPLES
137
10.1
LINEAR INDEPENDENCE
OF
MONOMORPHISMS
..............
137
11
FIELD
AUTOMORPHISMS
145
11.1 K-MONOMORPHISMS
..........
................
145
11.2 NORMAL CLOSURES
............
...............
146
12
THE GAELOIS CORRESPONDENCE
151
12.1
THE FUNDAMENTAL THEOREM
OF
GALOIS THEORY
........ ....
151
13
A WORKED EXAMPLE
155
CONTENTS
IX
14
SOLUBILITY
AND
SIMPLICITY
161
14.1 SOLUBLE GROUPS
........................
....
161
14.2 SIMPLE GROUPS
............
.... ..... . ......
164
14.3 CAUCHY S THEOREM
.... ................
....
166
15
SOLUTION
BY RADICALS
171
15.1 RADICAL EXTENSIONS
....... ........ ...........
171
15.2 AN INSOLUBLE QUINTIC
..... ..... .... ...........
176
15.3 OTHER METHODS
........
......... . ..........
178
16
ABSTRACT
RINGS
AND
FIELDS
181
16.1
RINGS
AND
FIELDS
...........................
181
16.2 GENERAL PROPERTIES
OF
RINGS
AND
FIELDS
.............
..
184
16.3
POLYNOMIALS
OVER GENERAL RINGS
............. .
....
186
16.4
THE CHARACTERISTIC
OF A
FIELD
...................
..
187
16.5
INTEGRAL DOMAINS
. ... .... ...................
188
17
ABSTRACT
FIELD
EXTENSIONS
193
17.1
MINIMAL POLYNOMIALS
.........
..... .... .......
193
17.2 SIMPLE
ALGEBRAIC
EXTENSIONS
......
.... .... .......
194
17.3
SPLITTING FIELDS
...... ... ....
.... .... .......
195
17.4
NORMALITY
..............
.................
197
17.5
SEPARABILITY
.... ....
... ...............
....
197
17.6
GALOIS THEORY FOR ABSTRACT
FIELDS
... ...............
202
18
THE
GENERAL
POLYNOMIAL
EQUATION
205
18.1
TRANSCENDENCE
DEGREE
.......
.... ...... ... ....
205
18.2
ELEMENTARY SYMMETRIC POLYNOMIALS
.................
208
18.3
THE GENERAL POLYNOMIAL
......................
209
18.4
CYCLIC EXTENSIONS
....
. .....................
211
18.5
SOLVING EQUATIONS
OF
DEGREE FOUR
OR
LESS
............
..
214
19
FINITE
FIELDS
221
19.1
STRUCTURE
OF
FINITE
FIELDS
.
...............
.......
221
19.2
THE MULTIPLICATIVE
GROUP
..................
....
222
19.3
APPLICATION
TO
SOLITAIRE
........................
224
20
REGULAR
POLYGONS
227
20.1
WHAT
EUCLID KNEW
...............
.
...
..... ..
227
20.2
WHICH CONSTRUCTIONS
ARE
POSSIBLE?
.................
230
20.3
REGULAR POLYGONS
...........................
231
20.4
FERMAT
NUMBERS
...........................
235
20.5
HOW
TO
DRAW
A
REGULAR 17-GON
...................
235
X
CONTENTS
21
CIRCLE DIVISION
243
21.1
GENUINE
RADICALS
...........................
244
21.2
FIFTH ROOTS REVISITED
...... .... ...
.
.......
....
246
21.3
VANDERMONDE REVISITED
.......................
249
21.4
THE GENERAL CASE
.
....
...
. .......... .
... ....
250
21.5
CYCLOTOMIC POLYNOMIALS
.......................
253
21.6
GALOIS
GROUP
OF
Q(Y)
:Q......................
255
21.7 THE TECHNICAL
LEMMA
........................
256
21.8 MORE
ON
CYCLOTOMIC
POLYNOMIALS
..................
257
21.9 CONSTRUCTIONS
USING
A
TRISECTOR
...................
259
22 CALCULATING
GALOIS
GROUPS
267
22.1 TRANSITIVE
SUBGROUPS
.........................
267
22.2 BARE
HANDS
ON
THE
CUBIC
.......................
268
22.3 THE DISCRIMINANT
...........................
271
22.4 GENERAL
ALGORITHM FOR
THE
GALOIS
GROUP
....
....... ....
272
23 ALGEBRAICALLY
CLOSED FIELDS
277
23.1 ORDERED FIELDS
AND
THEIR EXTENSIONS
................
277
23.2 SYLOW S
THEOREM
.
... .... .... ...............
279
23.3 THE ALGEBRAIC PROOF
.........................
281
24
TRANSCENDENTAL
NUMBERS
285
24.1 IRRATIONALITY
..............................
286
24.2
TRANSCENDENCE
OF E
... .......................
288
24.3 TRANSCENDENCE
OF
7R
.........................
289
25 WHAT DID GALOIS DO
OR
KNOW?
295
25.1 LIST
OF
THE
RELEVANT
MATERIAL
.....................
296
25.2 THE FIRST MEMOIR
..........................
296
25.3
WHAT GALOIS PROVED
... .....
......... . ... ... .
297
25.4 WHAT IS GALOIS
UP TO?
..... ... .....
... . ... ....
299
25.5 ALTERNATING GROUPS,
ESPECIALLY A5
..................
301
25.6 SIMPLE
GROUPS
KNOWN
TO
GALOIS
...................
302
25.7 SPECULATIONS
ABOUT
PROOFS
......................
303
REFERENCES
309
INDEX
.
315
|
| any_adam_object | 1 |
| author | Stewart, Ian 1945- |
| author_GND | (DE-588)124091598 |
| author_facet | Stewart, Ian 1945- |
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| classification_rvk | SK 230 |
| ctrlnum | (OCoLC)935827876 (DE-599)BVBBV043482129 |
| dewey-full | 512/.32 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.32 |
| dewey-search | 512/.32 |
| dewey-sort | 3512 232 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Fourth edition |
| format | Book |
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| genre | Lehrbuch gnd |
| genre_facet | Lehrbuch |
| id | DE-604.BV043482129 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:26:54Z |
| institution | BVB |
| isbn | 9781482245820 |
| language | English |
| lccn | 015458404 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028898867 |
| oclc_num | 935827876 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-83 DE-210 |
| owner_facet | DE-19 DE-BY-UBM DE-83 DE-210 |
| physical | xxii, 321 Seiten Illustrationen, Diagramme 24 cm |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | CRC Press, Taylor & Francis Group |
| record_format | marc |
| series2 | A Chapman & Hall book |
| spelling | Stewart, Ian 1945- (DE-588)124091598 aut Galois theory Ian Stewart, University of Warwick, Coventry, UK Fourth edition Boca Raton ; London ; New York CRC Press, Taylor & Francis Group [2015] © 2015 xxii, 321 Seiten Illustrationen, Diagramme 24 cm txt rdacontent n rdamedia nc rdacarrier A Chapman & Hall book Previous edition, 2004. - Includes bibliographical references (pages 309-314) and index Galois theory Algebraische Gleichung (DE-588)4001162-8 gnd rswk-swf Galois-Theorie (DE-588)4155901-0 gnd rswk-swf Lehrbuch gnd rswk-swf Galois-Theorie (DE-588)4155901-0 s Lehrbuch f DE-604 Algebraische Gleichung (DE-588)4001162-8 s 1\p DE-604 Digitalisierung Deutsches Museum application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028898867&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Stewart, Ian 1945- Galois theory Galois theory Algebraische Gleichung (DE-588)4001162-8 gnd Galois-Theorie (DE-588)4155901-0 gnd |
| subject_GND | (DE-588)4001162-8 (DE-588)4155901-0 |
| title | Galois theory |
| title_auth | Galois theory |
| title_exact_search | Galois theory |
| title_full | Galois theory Ian Stewart, University of Warwick, Coventry, UK |
| title_fullStr | Galois theory Ian Stewart, University of Warwick, Coventry, UK |
| title_full_unstemmed | Galois theory Ian Stewart, University of Warwick, Coventry, UK |
| title_short | Galois theory |
| title_sort | galois theory |
| topic | Galois theory Algebraische Gleichung (DE-588)4001162-8 gnd Galois-Theorie (DE-588)4155901-0 gnd |
| topic_facet | Galois theory Algebraische Gleichung Galois-Theorie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028898867&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT stewartian galoistheory |