Galois theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
New York, NNY [u.a.]
Springer
2001
|
| Series: | Graduate Texts in Mathematics
204 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 271 - 276 |
| Physical Description: | XIV, 280 Seiten graph. Darst. |
| ISBN: | 0387987657 |
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MARC
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| 100 | 1 | |a Escofier, Jean-Pierre |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Théorie de Galois |
| 245 | 1 | 0 | |a Galois theory |c Jean-Pierre Escofier |
| 264 | 1 | |a New York, NNY [u.a.] |b Springer |c 2001 | |
| 300 | |a XIV, 280 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate Texts in Mathematics |v 204 | |
| 500 | |a Literaturverz. S. 271 - 276 | ||
| 650 | 4 | |a Galois-Theorie | |
| 650 | 4 | |a Galois theory | |
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Record in the Search Index
| _version_ | 1826212312177967104 |
|---|---|
| adam_text |
JEAN-PIERRE ESCOFIER
GALOIS THEORY
TRANSLATED BY LEILA SCHNEPS
WITH 48 ILLUSTRATIONS
SPRINGER
CONTENTS
PREFACE V
1 HISTORICAL ASPECTS OF THE RESOLUTION OF ALGEBRAIC
EQUATIONS
1
1.1 APPROXIMATING THE ROOTS OF AN EQUATION 1
1.2 CONSTRUCTION OF SOLUTIONS BY INTERSECTIONS OF CURVES .
.
.
. 2
1.3
RELATIONS WITH TRIGONOMETRY 2
1.4 PROBLEMS OF NOTATION AND TERMINOLOGY 3
1.5 THE PROBLEM OF LOCALIZATION OF THE ROOTS 4
1.6 THE PROBLEM OF THE EXISTENCE OF ROOTS 5
1.7 THE PROBLEM OF ALGEBRAIC SOLUTIONS OF EQUATIONS 6
'" TOWARD CHAPTER 2 7
2 RESOLUTION OF QUADRATIC, CUBIC, AND QUARTIC EQUATIONS 9
2.1
SECOND-DEGREE EQUATIONS 9
2.1.1 THE BABYLONIANS 9
2.1.2 THE GREEKS . 11
2.1.3 THE ARABS 11
2.1.4 USE OF NEGATIVE NUMBERS 12
2.2 CUBIC EQUATIONS 13
2.2.1 THE GREEKS 13
2.2.2 OMAR KHAYYAM AND SHARAF AD DIN AT TUSI 13
2.2.3 SCIPIO DEL FERRO, TARTAGLIA, CARDAN 14
2.2.4 ALGEBRAIC SOLUTION OF THE CUBIC EQUATION 15
2.2.5 FIRST COMPUTATIONS WITH COMPLEX NUMBERS 16
2.2.6 RAIFAELE BOMBELLI 17
VIII CONTENTS
2.2.7 FRANCOIS VIETE 18
2.3 QUARTIC EQUATIONS 18
EXERCISES FOR CHAPTER 2 19
SOLUTIONS TO SOME OF THE EXERCISES 22
3 SYMMETRIC POLYNOMIALS 25
3.1 SYMMETRIC POLYNOMIALS 25
3.1.1 BACKGROUND 25
3.1.2 DEFINITIONS 26
3.2 ELEMENTARY SYMMETRIC POLYNOMIALS 27
3.2.1 DEFINITION 27
3.2.2 THE PRODUCT OF THE
X
- X*; RELATIONS BETWEEN CO
EFFICIENTS
AND ROOTS 27
3.3 SYMMETRIC POLYNOMIALS AND ELEMENTARY SYMMETRIC POLYNO
MIALS
29
3.3.1 THEOREM 29
3.3.2 PROPOSITION 31
3.3.3 PROPOSITION 32
3.4 NEWTON'S FORMULAS 32
3.5 RESULTANT OF TWO POLYNOMIALS 35
3.5.1 DEFINITION 35
3.5.2 PROPOSITION 35
3.6 DISCRIMINANT OF A POLYNOMIAL 37
3.6.1 DEFINITION 37
3.6.2 PROPOSITION 37
3.6.3 FORMULAS 38
3.6.4 POLYNOMIALS WITH REAL COEFFICIENTS: REAL ROOTS AND
SIGN
OF THE DISCRIMINANT 38
EXERCISES FOR CHAPTER 3 39
SOLUTIONS TO SOME OF THE EXERCISES 44
4 FIELD EXTENSIONS 51
4.1 FIELD EXTENSIONS 51
4.1.1 DEFINITION 51
4.1.2 PROPOSITION 52
4.1.3 THE DEGREE OF AN EXTENSION 52
4.1.4 TOWERS OF FIELDS 52
4.2 THE TOWER RULE - . . . 53
4.2.1 PROPOSITION 53
4.3 GENERATED EXTENSIONS 54
4.3.1 PROPOSITION 54
4.3.2 DEFINITION". 55
4.3.3 PROPOSITION 55
4.4 ALGEBRAIC ELEMENTS 55
4.4.1 DEFINITION 55
CONTENTS
IX
4.4.2 TRANSCENDENTAL NUMBERS
55
4.4.3 MINIMAL POLYNOMIAL
OF AN
ALGEBRAIC ELEMENT
.
.
.
. 56
4.4.4
DEFINITION
56
4.4.5
_
PROPERTIES
OF THE
MINIMAL POLYNOMIAL
57
4.4.6 PROVING
THE
IRREDUCIBILITY
OF A
POLYNOMIAL
IN
Z[X] .
57
4.5
ALGEBRAIC EXTENSIONS
59
4.5.1 EXTENSIONS GENERATED
BY AN
ALGEBRAIC ELEMENT
.
.
. 59
4.5.2
PROPERTIES
OF
IF [A]
59
4.5.3 DEFINITION
60
4.5.4 EXTENSIONS
OF
FINITE DEGREE
60
4.5.5 COROLLARY: TOWERS
OF
ALGEBRAIC EXTENSIONS
61
4.6 ALGEBRAIC EXTENSIONS GENERATED
BY
N
ELEMENTS
61
4.6.1
NOTATION
61
4.6.2 PROPOSITION
61
4.6.3 COROLLARY
62
4.7 CONSTRUCTION
OF AN
EXTENSION
BY
ADJOINING
A
ROOT
62
4.7.1 DEFINITION
62
4.7.2 PROPOSITION
62
4.7.3 COROLLARY
63
4.7.4 UNIVERSAL PROPERTY
OF
K[X]/(P)
63
TOWARD
CHAPTERS
5 AND 6 64
EXERCISES
FOR
CHAPTER
4 64
SOLUTIONS
TO
SOME
OF THE
EXERCISES
69
CONSTRUCTIONS WITH STRAIGHTEDGE AND COMPASS 79
5.1
CONSTRUCTIBLE POINTS 79
5.2 EXAMPLES OF CLASSICAL CONSTRUCTIONS 80
5.2.1 PROJECTION OF A POINT ONTO A LINE 80
5.2.2 CONSTRUCTION OF AN ORTHONORMAL BASIS FROM TWO POINTS 80
5.2.3
CONSTRUCTION OF A LINE PARALLEL TO A GIVEN LINE PASS
ING THROUGH A POINT . 81
5.3 LEMMA 82
5.4 COORDINATES OF POINTS CONSTRUCTIBLE IN ONE STEP 82
5.5
A NECESSARY CONDITION FOR CONSTRUCTIBILITY
[YYYYYYYY
83
5.6
TWO
PROBLEMS MORE THAN
TWO
THOUSAND YEARS
OLD .
.
.
. 84
5.6.1
DUPLICATION
OF THE
CUBE
85
5.6.2 TRISECTION
OF THE
ANGLE
85
5.7
A
SUFFICIENT CONDITION
FOR
CONSTRUCTIBILITY
85
EXERCISES
FOR
CHAPTER
5 87
SOLUTIONS
TO
SOME
OF THE
EXERCISES
90
IF-HOMOMORPHISMS
^ 93
6.1 CONJUGATE NUMBERS
93
6.2 IF-HOMOMORPHISMS
94
6.2.1 DEFINITIONS
94
CONTENTS
6.2.2 PROPERTIES 94
6.3 ALGEBRAIC ELEMENTS AND IF-HOMOMORPHISMS 95
6.3.1 PROPOSITION 95
6.3.2 EXAMPLE 96
6.4 EXTENSIONS OF EMBEDDINGS INTO C 97
6.4.1 DEFINITION 97
6.4.2 PROPOSITION 97
6.4.3 PROPOSITION 98
6.5 THE PRIMITIVE ELEMENT THEOREM 99
6.5.1 THEOREM AND DEFINITION 99
6.5.2 EXAMPLE 100
6.6 LINEAR INDEPENDENCE OF IF-HOMOMORPHISMS 101
6.6.1 CHARACTERS 101
6.6.2 EMIL ARTIN'S THEOREM 101
6.6.3 COROLLARY: DEDEKIND'S THEOREM 102
EXERCISES FOR CHAPTER 6 102
SOLUTIONS TO SOME OF THE EXERCISES " 103
NORMAL EXTENSIONS 107
7.1 SPLITTING FIELDS 107
7.1.1 DEFINITION 107
7.1.2 SPLITTING FIELD OF A CUBIC POLYNOMIAL 108
7.2 NORMAL EXTENSIONS 108
7.3 NORMAL EXTENSIONS AND IF-HOMOMORPHISMS 109
7.4 SPLITTING FIELDS AND NORMAL EXTENSIONS 109
7.4.1 PROPOSITION 109
7.4.2 CONVERSE 110
7.5 NORMAL EXTENSIONS AND INTERMEDIATE EXTENSIONS 110
7.6
NORMAL CLOSURE ILL
7.6.1 DEFINITION ILL
7.6.2 PROPOSITION ILL
7.6.3 PROPOSITION ILL
7.7 SPLITTING FIELDS: GENERAL CASE 112
TOWARD CHAPTER 8 113
EXERCISES FOR CHAPTER 7 113
SOLUTIONS TO SOME OF THE EXERCISES 115
GALOIS GROUPS
119
8.1 GALOIS GROUPS 119
8.1.1 THE GALOIS GROUP OF AN EXTENSION 119
8.1.2 THE ORDER OF THE GALOIS GROUP OF A NORMAL EXTEN
SION
OF-FINITE DEGREE 120
8.1.3 THE GALOIS GROUP OF A POLYNOMIAL 120
8.1.4 THE GALOIS GROUP AS A SUBGROUP OF A PERMUTATION
GROUP
120
CONTENTS XI
8.1.5 A SHORT HISTORY OF GROUPS 121
-8.2 FIELDS OF INVARIANTS 122
8.2.1 DEFINITION AND PROPOSITION 122
8.2.2^ EMIL ARTIN'S THEOREM 122
8.3 THE^EXAMPLE OF Q
[V^,J]:
FIRST PART 124
8.4 GALOIS GROUPS AND INTERMEDIATE EXTENSIONS 126
8.5 THE GALOIS CORRESPONDENCE 126
8.6 THE EXAMPLE OF Q
[\/2,J]:
SECOND PART 128
8.7
THE EXAMPLE X
4
+ 2 128
8.7.1 DIHEDRAL GROUPS 128
8.7.2 THE SPECIAL CASE OF
D
A
129
8.7.3
THE GALOIS GROUP OF
X
4
+
2 130
8.7.4 THE GALOIS CORRESPONDENCE . 130
8.7.5 SEARCH FOR MINIMAL POLYNOMIALS 132
TOWARD CHAPTERS 9, 10, AND 12 133
EXERCISES FOR CHAPTER 8 133
SOLUTIONS TO SOME OF THE EXERCISES 139
9 ROOTS OF UNITY 149
9.1 THE GROUP
U(N)
OF UNITS OF THE RING Z/NZ 149
9.1.1
DEFINITION AND BACKGROUND 149
9.1.2 THE STRUCTURE OF
U(N)
150
9.2
THE MOBIUS FUNCTION 151
9.2.1 MULTIPLICATIVE FUNCTIONS 151
9.2.2 THE MOBIUS FUNCTION 151
9.2.3 PROPOSITION 151
9.2.4 THE MOBIUS INVERSION FORMULA 152
9.3 ROOTS OF UNITY 153
9.3.1 N-TH ROOTS OF UNITY 153
9.3.2 PROPOSITION 153
._
9.3.3 PRIMITIVE ROOTS 153
9.3.4 PROPERTIES OF PRIMITIVE ROOTS 153
9.4 CYCLOTOMIC POLYNOMIALS 153
9.4.1 DEFINITION 153
9.4.2 PROPERTIES OF THE CYCLOTOMIC POLYNOMIAL 153
9.5
THE GALOIS GROUP OVER Q OF AN EXTENSION OF Q BY A ROOT
OF
UNITY 156
EXERCISES FOR CHAPTER 9 157
SOLUTIONS TO SOME OF THE EXERCISES 163
10 CYCLIC EXTENSIONS 179
10.1 CYCLIC AND ABELIAN EXTENSIONS 179
10.2 EXTENSIONS BY A ROOT AND CYCLIC EXTENSIONS 179
10.3 IRREDUCIBILITY OF
X
P
-
A
180
10.4 HILBERT'S THEOREM 90 181
XII CONTENTS
10.4.1 THE NORM 181
10.4.2 HILBERT'S THEOREM 90 182
10.5 EXTENSIONS BY A ROOT AND CYCLIC EXTENSIONS: CONVERSE . . . 182
10.6
LAGRANGE RESOLVENTS 183
10.6.1 DEFINITION 183
10.6.2 PROPERTIES 183
10.7 RESOLUTION OF THE CUBIC EQUATION 184
10.8 SOLUTION OF THE QUARTIC EQUATION 186
10.9 HISTORICAL COMMENTARY 188
EXERCISES FOR CHAPTER 10 188
SOLUTIONS TO SOME OF THE EXERCISES 190
11 SOLVABLE GROUPS 195
11.1 FIRST DEFINITION 195
11.2 DERIVED OR COMMUTATOR SUBGROUP 196
11.3 SECOND DEFINITION OF SOLVABILITY 196
11.4 EXAMPLES OF SOLVABLE GROUPS 197
11.5 THIRD DEFINITION 197
11.6 THE GROUP
A
N
IS SIMPLE FOR
N
5 198
11.6.1
THEOREM 198
11.6.2
AN
IS NOT SOLVABLE FOR N 5, DIRECT PROOF 199
11.7
RECENT RESULTS
." 199
EXERCISES FOR CHAPTER 11 200
SOLUTIONS TO SOME OF THE EXERCISES 203
12 SOLVABILITY OF EQUATIONS BY RADICALS 207
12.1 RADICAL EXTENSIONS AND POLYNOMIALS SOLVABLE BY RADICALS . 207
12.1.1
RADICAL EXTENSIONS 207
12.1.2 POLYNOMIALS SOLVABLE BY RADICALS 208
12.1.3 FIRST CONSTRUCTION 208
12.1.4 SECOND CONSTRUCTION 208
12.2 IF A POLYNOMIAL IS SOLVABLE BY RADICALS, ITS GALOIS GROUP IS
SOLVABLE
209
12.3 EXAMPLE OF A POLYNOMIAL NOT SOLVABLE BY RADICALS 209
12.4
THE CONVERSE OF THE FUNDAMENTAL CRITERION 210
12.5
THE GENERAL EQUATION OF DEGREE
N
210
12.5.1
ALGEBRAICALLY INDEPENDENT ELEMENTS . . . 210
12.5.2 EXISTENCE OF ALGEBRAICALLY INDEPENDENT ELEMENTS . . 211
12.5.3
THE GENERAL EQUATION OF DEGREE N 211
12.5.4 GALOIS GROUP OF THE GENERAL EQUATION OF DEGREE
N .
211
EXERCISES
FOR CHAPTER 12 . . .' 212
SOLUTIONS TO SOME OF THE EXERCISES 214
13 THE LIFE OF EVARISTE GALOIS 219
CONTENTS XIII
14 FINITE FIELDS 227
X
14.1 ALGEBRAICALLY CLOSED FIELDS 227
14.1.1 DEFINITION 227
14.1.2 ALGEBRAIC CLOSURES 228
14.1.3 THEOREM (STEINITZ, 1910) 228
14.2 EXAMPLES OF FINITE FIELDS 229
14.3 THE CHARACTERISTIC OF A FIELD 229
14.3.1 DEFINITION 229
14.3.2 PROPERTIES 229
14.4 PROPERTIES OF FINITE FIELDS 230
14.4.1 PROPOSITION 230
14.4.2 THE FROBENIUS HOMOMORPHISM 231
14.5 EXISTENCE AND UNIQUENESS OF A FINITE FIELD WITH
P
R
ELEMENTS 231
14.5.1 PROPOSITION 231
14.5.2 COROLLARY 232
14.6 EXTENSIONS OF FINITE FIELDS 233
14.7 NORMALITY OF A FINITE EXTENSION OF FINITE FIELDS 233
14.8
THE GALOIS GROUP OF A FINITE EXTENSION OF A FINITE FIELD . . 233
14.8.1
PROPOSITION 233
14.8.2 THE GALOIS CORRESPONDENCE 234
14.8.3 EXAMPLE 234
EXERCISES FOR CHAPTER 14 235
SOLUTIONS TO SOME OF THE EXERCISES 243
15 SEPARABLE EXTENSIONS 257
15.1 SEPARABILITY 257
15.2 EXAMPLE OF AN INSEPARABLE ELEMENT 258
15.3 A CRITERION FOR SEPARABILITY 258
15.4 PERFECT FIELDS 259
15.5 PERFECT FIELDS AND SEPARABLE EXTENSIONS 259
15.6 GALOIS EXTENSIONS 260
15.6.1 DEFINITION 260
15.6.2 PROPOSITION 260
15.6.3 THE GALOIS CORRESPONDENCE 260
TOWARD CHAPTER 16 ; 260
16 RECENT DEVELOPMENTS 261
16.1 THE INVERSE PROBLEM OF GALOIS THEORY 261
16.1.1 THE PROBLEM 261
16.1.2 THE ABELIAN CASE 262
16.1.3 EXAMPLE; 262
16.2 COMPUTATION OF
GALOIS
GROUPS OVER Q FOR SMALL-DEGREE POLY
NOMIALS
262
16.2.1 SIMPLIFICATION OF THE PROBLEM 263
16.2.2 THE IRREDUCIBILITY PROBLEM 263
XIV CONTENTS
16.2.3 EMBEDDING OF
G
INTO
S
N
263
16.2.4
LOOKING FOR
G
AMONG THE TRANSITIVE SUBGROUPS OF
S
N
264
16.2.5
TRANSITIVE SUBGROUPS OF 5
4
264
.16.2.6
STUDY OF $(G)
C A
N
265
16.2.7
STUDY OF $(G) C
D
4
266
16.2.8
STUDY OF $(G) C Z/4Z 267
16.2.9 AN ALGORITHM FOR N"= 4 268
BIBLIOGRAPHY 271
INDEX 277 |
| any_adam_object | 1 |
| author | Escofier, Jean-Pierre |
| author_facet | Escofier, Jean-Pierre |
| author_role | aut |
| author_sort | Escofier, Jean-Pierre |
| author_variant | j p e jpe |
| building | Verbundindex |
| bvnumber | BV013646218 |
| callnumber-first | Q - Science |
| callnumber-label | QA174 |
| callnumber-raw | QA174.2 |
| callnumber-search | QA174.2 |
| callnumber-sort | QA 3174.2 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 230 SK 200 |
| ctrlnum | (OCoLC)247403595 (DE-599)BVBBV013646218 |
| dewey-full | 512.3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.3 |
| dewey-search | 512.3 |
| dewey-sort | 3512.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre | 1\p (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV013646218 |
| illustrated | Illustrated |
| indexdate | 2025-03-10T13:02:57Z |
| institution | BVB |
| isbn | 0387987657 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009323524 |
| oclc_num | 247403595 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-20 DE-703 DE-19 DE-BY-UBM DE-706 DE-29T DE-634 DE-11 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-20 DE-703 DE-19 DE-BY-UBM DE-706 DE-29T DE-634 DE-11 DE-188 |
| physical | XIV, 280 Seiten graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Springer |
| record_format | marc |
| series | Graduate Texts in Mathematics |
| series2 | Graduate Texts in Mathematics |
| spelling | Escofier, Jean-Pierre Verfasser aut Théorie de Galois Galois theory Jean-Pierre Escofier New York, NNY [u.a.] Springer 2001 XIV, 280 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate Texts in Mathematics 204 Literaturverz. S. 271 - 276 Galois-Theorie Galois theory Galois-Theorie (DE-588)4155901-0 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Galois-Theorie (DE-588)4155901-0 s DE-604 Graduate Texts in Mathematics 204 (DE-604)BV000000067 204 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009323524&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 | Escofier, Jean-Pierre Galois theory Graduate Texts in Mathematics Galois-Theorie Galois theory Galois-Theorie (DE-588)4155901-0 gnd |
| subject_GND | (DE-588)4155901-0 (DE-588)4123623-3 |
| title | Galois theory |
| title_alt | Théorie de Galois |
| title_auth | Galois theory |
| title_exact_search | Galois theory |
| title_full | Galois theory Jean-Pierre Escofier |
| title_fullStr | Galois theory Jean-Pierre Escofier |
| title_full_unstemmed | Galois theory Jean-Pierre Escofier |
| title_short | Galois theory |
| title_sort | galois theory |
| topic | Galois-Theorie Galois theory Galois-Theorie (DE-588)4155901-0 gnd |
| topic_facet | Galois-Theorie Galois theory Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009323524&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT escofierjeanpierre theoriedegalois AT escofierjeanpierre galoistheory |