Topological Galois theory: solvability and unsolvability of equations in finite terms
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| Language: | English |
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Heidelberg ; New York ; Dordrecht ; London
Springer
[2014]
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| Series: | Springer monographs in mathematics
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| Subjects: | |
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| Physical Description: | xviii, 307 Seiten Illustrationen |
| ISBN: | 9783642388705 |
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| 100 | 1 | |a Chovanskij, Askolʹd G. |d 1947- |e Verfasser |0 (DE-588)1064596754 |4 aut | |
| 240 | 1 | 0 | |a Topologicheskaya Teoriya Galua |
| 245 | 1 | 0 | |a Topological Galois theory |b solvability and unsolvability of equations in finite terms |c Askold Khovanskii |
| 264 | 1 | |a Heidelberg ; New York ; Dordrecht ; London |b Springer |c [2014] | |
| 264 | 4 | |c © 2014 | |
| 300 | |a xviii, 307 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer monographs in mathematics | |
| 650 | 0 | 7 | |a Galois-Theorie |0 (DE-588)4155901-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Topologische Methode |0 (DE-588)4312758-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Galois-Theorie |0 (DE-588)4155901-0 |D s |
| 689 | 0 | 1 | |a Topologische Methode |0 (DE-588)4312758-7 |D s |
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| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-642-38871-2 |
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Contents
Construction
of LiouvilHan Classes of Functions
and Liouville's Theory
.
I
1.1
Defining Classes of Functions by Lists of Basic
Functions and Admissible Operations
. 2
1
.2
LiouvilHan Classes of Functions of a Single Variable
. 3
1.2.1
Functions of One Variable Representable
by Radicals
. 3
1.2.2
Elementary Functions of One Variable
. 5
1.2.3
Functions of One Variable Representable
by Quadrature
. 5
1.3
A Bit of History
. 6
1.4
New Definitions of Liouvillian Classes of Functions
. 7
1.4.1
Elementary Functions of One Variable
. 7
1.4.2
Functions of One Variable Representable
by Quadratures
. 8
1.4.3
Generalized Elementary Functions of One
Variable and Functions of One Variable
Representable by Generalized Quadratures
and A:-Quadratures
. 8
1.5
Liouville Extensions of Abstract and Functional
Differential Fields
. 10
1.6
Integration of Elementary Functions
. 13
.6.1
Liouville's Theorem: Outline of a Proof
. 15
6.2
Refinement of Liouville's Theorem
. 16
.6.3
Algebraic Extensions of Differential Fields
. 17
.6.4
Extensions of Transcendence Degree One
. 18
1.6.5
Adjunction of an Integral and an Exponential
of Integral
. 21
1.6.6
Proof of Liouville's Theorem
. 22
XI
xii Contents
1.7 Integration
of Functions Containing the Logarithm
. 25
1.7.1
The Polar Part of an Integral
. 25
1.7.2
The Logarithmic Derivative Part
. 26
1.7.3
Integration of a Polynomial of a Logarithm
. 27
1.7.4
Integration of Functions Lying in a Logarithmic
Extension of the Field €(z)
. 28
1.8
Integration of Functions Containing an Exponential
. 29
.8.1
Principal Polar Part of the Integral
. 29
.8.2
Principal Logarithmic Derivative Part
. 30
1
.8.3
Integration of Laurent Polynomials of the Exponential
_ 32
1.8.4 Solvability of First-Order Linear Differential Equations.
. 32
1.8.5
Integration of Functions Lying
in an Exponential Extension of the Field (D (z)
. 35
1.9
Integration of Algebraic Functions
. 35
1.9.1
The Rational Part of an Abelian Integral
. 36
1.9.2
Logarithmic Part of an Abelian Integral
. 38
1.9.3
Elementarily and Nonelementarity of Abelian Integrals
. 41
1.10
The Liouville-Mordukhai-Boltovski Criterion
. 44
Solvability of Algebraic Equations by Radicals and Galois Theory
— 47
2.1
Action of a Solvable Group and Representability by Radicals
. 49
2.1.1
A Sufficient Condition for Solvability by Radicals
. 49
2.1.2
The Permutation Group of the Variables
and Equations of Degree
2, 3,
and
4 . 51
2.1.3 Lagrange
Polynomials and Abelian
Linear-Algebraic Groups
. 52
2.1.4
Solving Equations of Degrees
2, 3,
and
4
by Radicals
. 55
2.2
Fixed Points of Finite Group Actions
. 58
2.3
Field Automorphisms and Relations Between Elements
in a Field
. 61
2.3.1
Equations Without Multiple Roots
. 61
2.3.2
Algebraicky
over an Invariant Subfield
. 61
2.3.3
Subalgebras Containing the Coefficients
of a Lagrange Polynomial
. 62
2.3.4
Representability of One Element Through
Another Element over an Invariant Subfield
. 63
2.4
Action of a fc-Solvable Group and Representability
by
к
-Radicals
. 64
2.5
Galois Equations
. 65
2.6
Automorphisms Related to a Galois Equation
. 67
2.7
The Fundamental Theorem of Galois Theory
. 68
2.7.1
Galois Extensions
. 68
2.7.2
Galois Groups
. 69
2.7.3
The Fundamental Theorem
. 70
2.7.4
Properties of the Galois Correspondence
. 70
2.7.5
Changing the Field of Coefficients
. 72
Contents xiii
2.8
A Criterion for Solvability of Equations by Radicals
. 73
2.8.1
Roots of Unity
. 73
2.8.2
The Equation x"
=
a
. 74
2.8.3
Solvability by Radicals
. 75
2.9
A Criterion for Solvability by
/с
-Radicals.
76
2.9.1
Properties of
/с
-Solvable Groups
. 76
2.9.2
Solvability by
к
-Radicals
. 78
2.9.3
Unsolvability of the General Equation
of Degree
к
+ 1 > 4
by Ar-Radicals
. 79
2.10
Unsolvability of Complicated Equations by Solving
Simpler Equations
. 81
2.10.1
A Necessary Condition for Solvability
. 81
2.
1
0.2
Classes of Finite Groups
. 82
3
Solvability and Picard-Vessiot Theory
. 85
3.1
Similarity Between Linear Differential Equations
and Algebraic Equations
. 85
3.1.1
Division with Remainder and the Greatest
Common Divisor of Differential Operators
. 85
3.1.2
Reduction of Order for a Linear Differential
Equation as an Analogue of
Bézouťs
Theorem
. 86
3.1.3
A Generic Linear Differential Equation
with Constant Coefficients and
Lagrange
Resolvents
. 87
3.1.4
Analogue of Viete's Formulas for Differential
Operators
. 88
3.1.5
An Analogue of the Theorem on Symmetric
Functions for Differential Operators
. 90
3.2
A Picard-Vessiot Extension and Its Galois Group
. 91
3.3
The Fundamental Theorem of Picard-Vessiot Theory
. 93
3.4
The Simplest Picard-Vessiot Extensions
. 94
3.4.1
Algebraic Extensions
. 94
3.4.2
Adjoining an Integral
. 95
3.4.3
Adjoining an Exponential of Integral
. 96
3.5
Solvability of Differential Equations
. 98
3.6
Linear Algebraic Groups and Necessary Conditions
of Solvability
. 99
3.7
A Sufficient Condition for the Solvability of Differential
Equations
. 101
3.8
Other Kinds of Solvability
. 104
4
Coverings and Galois Theory
. 107
4.1
Coverings over Topological Spaces
. 109
4.1.1
Classification of Coverings with Marked Points
. 109
4.1.2
Coverings with Marked Points and Subgroups
of the Fundamental Group
.
Ill
4.1.3
Other Classifications of Coverings
. 114
xiv
Contents
4.1.4
A Similarity Between Galois Theory
and the Classification of Coverings
. 117
4.2
Completion of Ramified Coverings and Riemann
Surfaces of Algebraic Functions
. 118
4.2.1
Filling Holes and Puiseux Expansions
. 119
4.2.2
Analytic-Type Maps and the Real Operation
of Filling Holes
. 121
4.2.3
Finite Ramified Coverings with a Fixed
Ramification Set
. 123
4.2.4
The Riemann Surface of an Algebraic Equation
over the Field of Meromorphic Functions
. 128
4.3
Finite Ramified Coverings and Algebraic Extensions
of Fields of Meromorphic Functions
. 130
4.3.1
The Field Pa(O) of Germs at the Point
а є
X
of Algebraic Functions with Ramification over
О
. 130
4.3.2
Galois Theory for the Action of the
Fundamental Group on the Field Pa(O)
. 132
4.3.3
Field of Functions on a Ramified Covering
. 134
4.4
Geometry of Galois Theory for Extensions of the Field
of Meromorphic Functions
. 136
4.4.1
Galois Extensions of the Field K(X)
. 136
4.4.2
Algebraic Extensions of the Field
of Germs of Meromorphic Functions
. 137
4.4.3
Algebraic Extensions of the Field of Rational
Functions
. 138
5
One-Dimensional
Topologicei
Galois Theory
. 143
5.1
On Topological Unsolvability
. 144
5.2
Topological Nonrepresentability of Functions by Radicals
. 147
5.2.1
Monodromy Groups of Basic Functions
. 148
5.2.2
Solvable Groups
. 149
5.2.3
The Class of Algebraic Functions with
Solvable Monodromy Groups Is Stable
. 149
5.2.4
An Algebraic Function with a Solvable
Monodromy Group Is Representable by Radicals
. 151
5.3
On the One-Dimensional Version of Topological Galois Theory
. 152
5.4
Functions with at Most Countable Singular Sets
. 153
5.4.
1 Forbidden Sets
. 154
5.4.2
The Class of ^-Functions Is Stable
. 155
5.5
Monodromy Groups
. 157
5.5.1
Monodromy Group with a Forbidden Set
. 157
5.5.2
Closed Monodromy Groups
. 158
5.5.3
Transitive Action of a Group on a Set
and the Monodromy Pair of an ^-Function
. 158
Contents xv
5.5.4 Almost Normal
Functions
. 159
5.5.5
Classes of Group Pairs
. 160
5.6
The Main Theorem
. 161
5.7
Group-Theoretic Obstructions to Representability
by Quadratures
. 164
5.7.1
Computation of Some Classes of Group Pairs
. 164
5.7.2
Necessary Conditions for Representability
by Quadratures,
к
-Quadratures,
and Generalized Quadratures
. 167
5.8
Classes of Singular Sets and a Generalization
of the Main Theorem
. ! 70
5.8.1
Functions Representable by Single-Valued
ΛΊ
-Functions and Quadratures
. 171
6
Solvability of Fuchsian Equations
. 173
6.1
Picard-
Vessiot Theory for Fuchsian Equations
. 173
6.1.1
The Monodromy Group of a Linear Differential
Equation and Its Connection with the Galois Group
. 173
6.1.2
Proof of Frobenius's Theorem
. 176
6.1.3
The Monodromy Group of Systems of Linear
Differential Equations and Its Connection with
the Galois Group
. 178
6.2
Galois Theory for Fuchsian Systems of Linear
Differential Equations with Small Coefficients
. 180
6.2.1
Fuchsian Systems of Equations
. 180
6.2.2
Groups Generated by Matrices Close to the Identity
. 182
6.2.3
Explicit Criteria for Solvability
. 185
6.2.4
Strong Unsol vability of Equations
. 187
6.3
Maps of the Half-Plane onto Polygons Bounded
by Circular Arcs
. 188
6.3.1
Using the Reflection Principle
. 188
6.3.2
Groups of Fractional Linear and
Conformai
Transformations of the Class ^{<C,
Jť)
. 189
6.3.3
Integrable
Cases
. 191
7
Multidimensional Topological Galois Theory
. 195
7.1
Introduction
. 195
7.1.1
Operations on Multivariate Functions
. 196
7.1.2
Liouvillian Classes of Multivariate Functions
. 197
7.1.3
New Definitions of Liouvillian Classes
of Multivariate Functions
. 200
7.1.4
Liouville Extensions of Differential Fields
Consisting of Multivariate Functions
. 202
xvi
Contents
7.2
Continuation of Multivalued Analytic Functions
to an Analytic Subset
. 204
7.2.1
Continuation of a Single-Valued Analytic
Function to an Analytic Subset
. 206
7.2.2
Admissible Stratifications
. 207
7.2.3
How the Topology of an Analytic Subset
Changes at an Irreducible Component
. 208
7.2.4
Covers Over the Complement of a Subset
of Hausdorff Codimension Greater Than
1
in a Manifold
. 210
7.2.5
Covers Over the Complement of an Analytic Set
. 213
7.2.6
The Main Theorem
. 215
7.3
On the Monodromy of a Multivalued Function
on Its Ramification Set
. 216
7.3.1
.^-Functions
. 217
7.3.2
Almost Homomorphisms and Induced Closures
. 219
7.3.3
Induced Closure of a Group Acting on a Set
in the Transformation Group of a Subset
. 221
7.3.4
The Monodromy Groups of Induced Functions
. 222
7.3.5
Classes of Group Pairs
. 224
7.4
Multidimensional Results on Nonrepresentability
of Functions by Quadratures
. 226
7.4.1
Formulas, Their Multigerms, Analytic
Continuations, and Riemann Surfaces
. 227
7.4.2
The Class of ^^-Germs, Its Stability Under
the Natural Operations
. 229
7.4.3
The Class of Formula Multigerms
with the .yV-Property
. 233
7.4.4
Topological Obstructions to Representability
of Functions by Quadratures
. 234
7.4.5
Monodromy Groups of Holonomic Systems
of Linear Differential Equations
. 236
7.4.6
Holonomic Systems of Linear Differential
Equations with Small Coefficients
. 237
A Straightedge and Compass Constructions
. 239
A.
1
Solvability of Equations by Square Roots
. 240
A.
1.1
Background Material
. 241
A.
1.2
Extensions by 2-Radicals
. 241
A.
1.3
2-Radical Extensions of a Field
of Characteristic
2. 243
A. 1
.4
Roots of Unity
. 243
A.
1.5
Solvability of the Equation xn
-
\
= 0
by 2-Radicals
. 245
A.
2
What Can Be Constructed Using Straightedge and Compass?
. 246
A.
2.
1 The Unsolvability of Some Straightedge and
Compass Construction Problems
. 247
Contents xvii
Α.
2.2
Some Explicit Constructions
. 248
A.
2.3
Classical Straightedge and Compass
Constructibility Problems
. 250
A.
2.4
Two Specific Constructions
. 251
A.2.5 Stratification of the Plane
. 252
A.
2.6
Classes of Constructions That Allow Arbitrary Choice
_ 253
A.
2.7
Trisection
of an Angle
. 254
A.
2.8
A Theorem from
Affine
Geometry
. 256
В
Chebyshev Polynomials and Their Inverses
. 257
B.
1
Chebyshev Functions over the Complex Numbers
. 258
B.
1.1
Multivalued Chebyshev Functions
. 258
B.
1.2
Germs of a Chebyshev Function at the Point
jc
- 1 . 260
B.
1.3
Analytic Continuation of Germs
. 26
1
B.2 Chebyshev Functions over Fields
. 262
B.2.
1
Algebraic Definition
. 262
B.2.
2
Equations of Degree Three
. 263
B.2.3 Equations of Degree Four
. 264
B.3 Three Classical Problems
. 265
B.3.
1
Inversion of Mappings in Radicals
. 265
B.3.
2
Inversion of Mappings of Finite Fields
. 267
B.3.3
Integrable
Mappings
. 268
С
Signatures of Branched Coverings and Solvability in Quadratures
_ 271
C.
1
Coverings with a Given Signature
. 272
C.
1.1
Definitions and Examples
. 272
C.
1.2
Classification
. 273
С
1.3
Coverings and Classical Geometries
. 274
C.2 The Spherical Case
. 276
C.2.I Application of the Riemann-Hurwitz Formula
. 276
C.2.
2
Finite Groups of Rotations of the Sphere
. 277
C.2.3 Coverings with Elliptic Signatures
. 278
C.2.4 Equations with an Elliptic Signature
. 278
C.3 The Case of the Plane
. 278
C.3.
1
Discrete Groups of
Affine
Transformations
. 278
C.3.
2
Affine
Groups Generated by Reflections
. 280
C.3.3 Coverings with Parabolic Signatures
. 280
C.3.4 Equations with Parabolic Signatures
. 281
C.4 Functions with Nonhyperbolic Signatures in Other Contexts
. 283
C.5 The Hyperbolic Case
. 284
xviii
Contents
D
On an Algebraic Version of Hubert's 13th Problem
. 287
D.I Versions of Hubert's
13thProblem
. 287
D.
1
Л
Simplification of Equations of High Degree
. 287
D.
1.2
Versions of the Problem for Different
Classes of Functions
. 288
D.2 Arnold's Theorem
. 289
D.2.
1
Formulation of the Theorem
. 289
D.2.2 Results Related to Arnold's Theorem
. 290
D.2.3 The Proof of the Theorem
. 291
D.2.4 Polynomial Versions of Klein's and Hubert's Problems
. 293
D.3 Klein's Problem
. 293
D.3.
1 Birational
Automorphisms and Klein's Problem
. 293
D.3.
2
Essential Dimension of Groups
. 295
D.3.3 A Topological Approach to Klein's Problem
. 296
D.4 Arnold's Proof and Further Developments in Klein's Problem
. 297
References
. 299
Index
. 305
Springer Monographs in Mathematics
Askold
Khovanskii
Topological
Galois Theory
Solvability and Unsol vability of Equations in Finite Terms
This book provides a detailed and largely self-contained description of various classical
and new results on solvability and unsolvability of equations in explicit form. In
particular, it offers a complete exposition of the relatively new area of topological Galois
theory, initiated by the author. Applications of Galois theory to solvability of algebraic
equations by radicals, basics of
Picard-
Vessiot theory, and Liouville's results on the
class of functions representable by quadratures are also discussed.
A unique feature of this book is that recent results are presented in the same elementary
manner as classical Galois theory, which will make the book useful and interesting
to readers with varied backgrounds in mathematics, from undergraduate students to
researchers.
In this English-language edition, extra material has been added (Appendices A-D),
the last two of which were written jointly with Yura
Burda. |
| any_adam_object | 1 |
| author | Chovanskij, Askolʹd G. 1947- |
| author_GND | (DE-588)1064596754 |
| author_facet | Chovanskij, Askolʹd G. 1947- |
| author_role | aut |
| author_sort | Chovanskij, Askolʹd G. 1947- |
| author_variant | a g c ag agc |
| building | Verbundindex |
| bvnumber | BV042176387 |
| classification_rvk | SK 200 SK 300 |
| ctrlnum | (OCoLC)864669034 (DE-599)DNB1034515195 |
| 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 | 3512.32 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042176387 |
| illustrated | Illustrated |
| indexdate | 2024-09-10T01:29:44Z |
| institution | BVB |
| isbn | 9783642388705 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027615717 |
| oclc_num | 864669034 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-188 |
| physical | xviii, 307 Seiten Illustrationen |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Chovanskij, Askolʹd G. 1947- Verfasser (DE-588)1064596754 aut Topologicheskaya Teoriya Galua Topological Galois theory solvability and unsolvability of equations in finite terms Askold Khovanskii Heidelberg ; New York ; Dordrecht ; London Springer [2014] © 2014 xviii, 307 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Galois-Theorie (DE-588)4155901-0 gnd rswk-swf Topologische Methode (DE-588)4312758-7 gnd rswk-swf Galois-Theorie (DE-588)4155901-0 s Topologische Methode (DE-588)4312758-7 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-38871-2 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4327748&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615717&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615717&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Chovanskij, Askolʹd G. 1947- Topological Galois theory solvability and unsolvability of equations in finite terms Galois-Theorie (DE-588)4155901-0 gnd Topologische Methode (DE-588)4312758-7 gnd |
| subject_GND | (DE-588)4155901-0 (DE-588)4312758-7 |
| title | Topological Galois theory solvability and unsolvability of equations in finite terms |
| title_alt | Topologicheskaya Teoriya Galua |
| title_auth | Topological Galois theory solvability and unsolvability of equations in finite terms |
| title_exact_search | Topological Galois theory solvability and unsolvability of equations in finite terms |
| title_full | Topological Galois theory solvability and unsolvability of equations in finite terms Askold Khovanskii |
| title_fullStr | Topological Galois theory solvability and unsolvability of equations in finite terms Askold Khovanskii |
| title_full_unstemmed | Topological Galois theory solvability and unsolvability of equations in finite terms Askold Khovanskii |
| title_short | Topological Galois theory |
| title_sort | topological galois theory solvability and unsolvability of equations in finite terms |
| title_sub | solvability and unsolvability of equations in finite terms |
| topic | Galois-Theorie (DE-588)4155901-0 gnd Topologische Methode (DE-588)4312758-7 gnd |
| topic_facet | Galois-Theorie Topologische Methode |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4327748&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615717&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=027615717&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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