Groups as Galois groups: an introduction
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| Format: | Book |
| Language: | Undetermined |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Pr.
1996
|
| Edition: | 1. publ. |
| Series: | Cambridge studies in advanced mathematics
53 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 248 S. graph. Darst. |
| ISBN: | 0521562805 |
Staff View
MARC
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| 245 | 1 | 0 | |a Groups as Galois groups |b an introduction |c Helmut Völklein |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Pr. |c 1996 | |
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| 490 | 1 | |a Cambridge studies in advanced mathematics |v 53 | |
| 650 | 4 | |a Galois-Gruppe | |
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Record in the Search Index
| _version_ | 1804125643052417024 |
|---|---|
| adam_text | Contents
Preface page xiii
Notation xvii
I The Basic Rigidity Criteria
1 Hilbert s Irreducibility Theorem 3
1.1 Hilbertian Fields 3
1.1.1 Preliminaries 3
1.1.2 Specializing the Coefficients of a Polynomial 5
1.1.3 Basic Properties of Hilbertian Fields 10
1.2 The Rational Field Is Hilbertian 13
1.2.1 Analyticity of Roots 14
1.2.2 The Rational Field Is Hilbertian 16
1.2.3 Integral Values of Meromorphic Functions 18
1.3 Algebraic Extensions of Hilbertian Fields 21
1.3.1 Weissauer s Theorem 21
1.3.2 Applications 24
2 Finite Galois Extensions of C(x) 26
2.1 Extensions of Laurent Series Fields 27
2.1.1 The Field of Formal Laurent Series over k 27
2.1.2 Factoring Polynomials over k[[t]] 28
2.1.3 The Finite Extensions of £((?)) 30
2.2 Extensions of k(x) 32
2.2.1 Branch Points and the Associated Conjugacy Classes 32
2.2.2 Riemann s Existence Theorem and Rigidity 37
3 Descent of Base Field and the Rigidity Criterion 40
3.1 Descent 40
3.1.1 Fields of Definition 40
3.1.2 The Descent from i to k 43
3.2 The Rigidity Criteria 48
vii
viii Contents
3.3 Rigidity and the Simple Groups 51
3.3.1 The Alternating and Symmetric Groups 52
3.3.2 A Formula to Verify Rigidity 53
3.3.3 The Sporadic Groups 54
3.3.4 The Lie Type Groups 55
3.3.5 A Criterion for Groups Modulo Center 56
3.3.6 An Example: The Groups PSL2O7) 57
4 Covering Spaces and the Fundamental Group 61
4.1 The General Theory 61
4.1.1 Homotopy 61
4.1.2 Coverings 63
4.1.3 The Homotopy Lifting Property 64
4.1.4 Galois Coverings and the Group of Deck
Transformations 67
4.2 Coverings of the Punctured Sphere 69
4.2.1 The Coverings of the Disc Minus Center 69
4.2.2 Coverings of the Punctured Sphere Behavior Near a
Ramified Point 72
4.2.3 Coverings of Prescribed Ramification Type 76
5 Riemann Surfaces and Their Function Fields 84
5.1 Riemann Surfaces 84
5.2 The Compact Riemann Surface Arising from a Covering of
the Punctured Sphere 87
5.2.1 Construction of an Atlas 87
5.2.2 The Identification between Topological and Algebraic
Ramification Type 89
5.3 Constructing Generators of G(L/C(x)) 92
5.4 Digression: The Equivalence between Coverings and Field
Extensions 94
6 The Analytic Version of Riemann s Existence Theorem 96
6.1 Abstract Hilbert Spaces 96
6.1.1 Continuous Linear Maps and Orthogonal
Complements 96
6.1.2 Banach s Theorem 99
6.2 The Hilbert Spaces L2(D) 100
6.2.1 Square Integrable Functions 101
6.2.2 Functions on a Disc 102
6.2.3 L2(D) Is a Hilbert Space 103
6.3 Cocycles and Coboundaries 105
Contents ix
6.3.1 Square Integrable Functions on Coordinate Patches 105
6.3.2 Cocycles 106
6.3.3 The Coboundary Map 107
6.4 Cocycles on a Disc 107
6.4.1 Dolbeault s Lemma 107
6.4.2 Cocycles on a Disc 109
6.5 A Finiteness Theorem 110
6.5.1 The Patching Process 111
6.5.2 Restriction Z (V) Z (U) 112
6.5.3 Proof of the Finiteness Theorem 114
II Further Directions
7 The Descent from C to k 119
7.1 Extensions of C(x) Unramified Outside a Given Finite Set 119
7.2 Specializing the Coefficients of an Absolutely Irreducible
Polynomial 121
7.3 The Descent from C to k 123
7.4 The Minimal Field of Definition 126
7.5 Embedding Problems over k(x) 128
8 Embedding Problems 130
8.1 Generalities 130
8.1.1 Fields over Which All Embedding Problems Are
Solvable 130
8.1.2 Minimal Embedding Problems 132
8.2 Wreath Products and Split Abelian Embedding Problems 134
8.2.1 A Rationality Criterion for Function Fields 134
8.2.2 The Group Theoretic Notion of Wreath Product 135
8.2.3 Wreath Products as Galois Groups 136
8.3 GAR Realizations and GAL Realizations 141
8.3.1 Definition and the Main Property of a GAR Realization 141
8.3.2 GAL Realizations 144
8.3.3 Digression: Fields of Cohomological Dimension 1 and
the Shafarevich Conjecture 149
8.3.4 GAL Realizations over k 153
9 Braiding Action and Weak Rigidity 155
9.1 Certain Galois Groups Associated with a Weakly Rigid
Ramification Type 156
9.2 Combinatorial Computation of A via Braid Group Action and
the Resulting Outer Rigidity Criterion 161
9.3 Construction of Weakly Rigid Tuples 165
x Contents
9.4 An Application of the Outer Rigidity Criterion 169
9.4.1 Braiding Action through the Matrices i (Q, f) 170
9.4.2 Galois Realizations for PGLn(q) and PUn(q) 173
10 Moduli Spaces for Covers of the Riemann Sphere 178
10.1 The Topological Construction of the Moduli Spaces 179
10.1.1 A Construction of Coverings 179
10.1.2 Distinguished Conjugacy Classes in the
Fundamental Group of a Punctured Sphere 181
10.1.3 The Moduli Spaces as Abstract Sets 181
10.1.4 The Topology of the Moduli Spaces 183
10.1.5 Families of Covers of the Riemann Sphere 186
10.1.6 The Braid Group 188
10.1.7 The Braiding Action on Generating Systems 192
10.1.8 Components of H .A)(G), and the Example of
Simple Covers 195
10.2 The Algebraic Structure of the Moduli Spaces 199
10.2.1 Coverings of Affine Varieties 200
10.2.2 The Action of Field Automorphisms on the
Points of ^ (G) 201
10.2.3 The Algebraic Structure of U(rA)(G), and the
Proof of Theorem 9.5 205
10.3 Digression: The Inverse Galois Problem and Rational Points
on Moduli Spaces 208
10.3.1 The Q Structure on H™{G) 208
10.3.2 Absolutely Irreducible Components of H rn(G)
Denned over Q 210
10.3.3 The Application to PAC Fields 212
11 Patching over Complete Valued Fields 213
11.1 Power Series over Complete Rings 214
11.1.1 Absolute Values 214
11.1.2 Power Series 215
11.1.3 Algebraic Power Series Are Convergent 216
11.1.4 WeierstraB Division 217
11.2 Rings of Converging Power Series 218
11.2.1 The Basic Set Up 219
11.2.2 Structure of the Rings A, AuandA2 220
11.2.3 The Embedding of A into k[[x c]] 222
11.3 GAGA 222
11.3.1 Cartan s Lemma 223
11.3.2 Induced Algebras 225
Contents xi
11.3.3 An Elementary Version of 1 Dimensional
Rigid GAGA 226
11.4 Galois Groups over k(x) 231
11.4.1 Inductive Construction of Galois Extensions 231
11.4.2 Regular Extensions in Positive Characteristic 235
11.4.3 Galois Realizations of Cyclic Groups 236
11.4.4 Galois Groups and Embedding Problems
over k(x) 239
References 243
Index 247
|
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| bvnumber | BV011149283 |
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| discipline | Mathematik |
| edition | 1. publ. |
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| indexdate | 2024-07-09T18:04:49Z |
| institution | BVB |
| isbn | 0521562805 |
| language | Undetermined |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007473301 |
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| physical | XVII, 248 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
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| publisher | Cambridge Univ. Pr. |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Völklein, Helmut 1957- Verfasser (DE-588)11016055X aut Groups as Galois groups an introduction Helmut Völklein 1. publ. Cambridge [u.a.] Cambridge Univ. Pr. 1996 XVII, 248 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 53 Galois-Gruppe Inverse Galois theory Galois-Gruppe (DE-588)4155897-2 gnd rswk-swf Galois-Gruppe (DE-588)4155897-2 s DE-604 Cambridge studies in advanced mathematics 53 (DE-604)BV000003678 53 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007473301&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Völklein, Helmut 1957- Groups as Galois groups an introduction Cambridge studies in advanced mathematics Galois-Gruppe Inverse Galois theory Galois-Gruppe (DE-588)4155897-2 gnd |
| subject_GND | (DE-588)4155897-2 |
| title | Groups as Galois groups an introduction |
| title_auth | Groups as Galois groups an introduction |
| title_exact_search | Groups as Galois groups an introduction |
| title_full | Groups as Galois groups an introduction Helmut Völklein |
| title_fullStr | Groups as Galois groups an introduction Helmut Völklein |
| title_full_unstemmed | Groups as Galois groups an introduction Helmut Völklein |
| title_short | Groups as Galois groups |
| title_sort | groups as galois groups an introduction |
| title_sub | an introduction |
| topic | Galois-Gruppe Inverse Galois theory Galois-Gruppe (DE-588)4155897-2 gnd |
| topic_facet | Galois-Gruppe Inverse Galois theory |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007473301&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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