Field arithmetic:
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Berlin
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[2008]
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| Ausgabe: | Third edition, revised by Moshe Jarden |
| Schriftenreihe: | Ergebnisse der Mathematik und ihrer Grenzgebiete
3. Folge ; volume 11 |
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| ISBN: | 9783540772699 |
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| 100 | 1 | |a Fried, Michael D. |d 1942- |e Verfasser |0 (DE-588)172081203 |4 aut | |
| 245 | 1 | 0 | |a Field arithmetic |c Michael D. Fried, Moshe Jarden |
| 250 | |a Third edition, revised by Moshe Jarden | ||
| 264 | 1 | |a Berlin |b Springer |c [2008] | |
| 264 | 4 | |c © 2008 | |
| 300 | |a xxiii, 792 Seiten |b Illustrationen |c 235 mm x 155 mm, 1310 gr. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |v volume 11 | |
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Datensatz im Suchindex
| _version_ | 1805090618663239680 |
|---|---|
| adam_text |
Table
of
Contents
Chapter
1.
Infinite Galois Theory and Profmite Groups
.1
1.1
Inverse Limits
.1
1.2
Profinite
Groups
.4
1.3
Infinite Galois Theory
.9
1.4
The p-adic Integers and the
Prüfer
Group
. 12
1.5
The Absolute Galois Group of a Finite Field
. 15
Exercises
. 16
Notes
. 18
Chapter
2.
Valuations and Linear Disjointness
. 19
2.1
Valuations, Places, and Valuation Rings
. 19
2.2
Discrete Valuations
. 21
2.3
Extensions of Valuations and Places
. 24
2.4
Integral Extensions and Dedekind Domains
. 30
2.5
Linear Disjointness of Fields
. 34
2.6
Separable, Regular, and Primary Extensions
. 38
2.7
The Imperfect Degree of a Field
. 44
2.8
Derivatives
. 48
Exercises
. 50
Notes
. 51
Chapter
3.
Algebraic Function Fields of One Variable
. 52
3.1
Function Fields of One Variable
. 52
3.2
The Riemann-Roch Theorem
. 54
3.3
Holomorphy Rings
. 56
3.4
Extensions of Function Fields
. 59
3.5
Completions
. 61
3.6
The Different
. 67
3.7
Hyperelliptic Fields
. 70
3.8
Hyperelliptic Fields with a Rational quadratic Subfield
. 73
Exercises
. 75
Notes
. 76
Chapter
4.
The Riemann Hypothesis for Function Fields
. 77
4.1
Class Numbers
. 77
4.2
Zeta
Functions
. 79
4.3
Zeta
Functions under Constant Field Extensions
. 81
4.4
The Functional Equation
. 82
4.5
The Riemann Hypothesis and Degree
1
Prime Divisors
. 84
4.6
Reduction Steps
. 86
4.7
An Upper Bound
. 87
4.8
A Lower Bound
. 89
viii Table of
Contents
Exercises .
91
Notes
. 93
Chapter
5.
Plane
Curves .
95
5.1
Affine
and
Projective
Plane Curves
. 95
5.2
Points and prime divisors
. 97
5.3
The Genus of a Plane Curve
. 99
5.4
Points on a Curve over a Finite Field
. 104
Exercises
. 105
Notes
. 106
Chapter
6.
The Chebotarev Density Theorem
. 107
6.1
Decomposition Groups
. 107
6.2
The Artin Symbol over Global Fields
.
Ill
6.3
Dirichlet Density
. 113
6.4
Function Fields
. 115
6.5
Number Fields
. 121
Exercises
. 129
Notes
. 130
Chapter
7.
Ultraproducts
. 132
7.1
First Order Predicate Calculus
. 132
7.2
Structures
. 134
7.3
Models
. 135
7.4
Elementary Substructures
. 137
7.5
Ultrafilters
. 138
7.6
Regular
Ultrafilters
. 139
7.7
Ultraproducts
. 141
7.8
Regular Ultraproducts
. 145
7.9
Xonprincipal Ultraproducts of Finite Fields
. 147
Exercises
. 147
Notes
. 148
Chapter
8.
Decision Procedures
. 149
8.1
Deduction Theory
. 149
8.2
Gödel's
Completeness Theorem
. 152
8.3
Primitive Recursive Functions
. 154
8.4
Primitive Recursive Relations
. 156
8.5
Recursive Functions
. 157
8.6
Recursive and Primitive Recursive Procedures
. 159
8.7
A Reduction Step in Decidability Procedures
. 160
Exercises
. 161
Notes
. 162
Table
of Contents
ix
Chapter
9.
Algebraically Closed Fields
. 163
9.1
Elimination of Quantifiers
. 163
9.2
A Quantifiers Elimination Procedure
. 165
9.3
Effectiveness
. 168
9.4
Applications
. . 169
Exercises
. 170
Notes
. 170
Chapter
10.
Elements of Algebraic Geometry
. 172
10.1
Algebraic Sets
. 172
10.2
Varieties
. 175
10.3
Substitutions in Irreducible Polynomials
. 176
10.4
Rational Maps
. 178
10.5 Hyperplane
Sections
. 180
10.6
Descent
. 182
10.7
Projective
Varieties
. 185
10.8
About the Language of Algebraic Geometry
. 187
Exercises
. 190
Notes
. 191
Chapter
11.
Pseudo
Algebraically Closed Fields
. 192
11.1
PAC
Fields
.". 192
11.2
Reduction to Plane Curves
. 193
11.3
The
PAC
Property is an Elementary Statement
. 199
11.4
PAC
Fields of Positive Characteristic
. 201
11.5
PAC
Fields with Valuations
. 203
11.6
The Absolute Galois Group of
a PAC
Field
. 207
11.7
A non-PAC
Field
К
with
ϋΓίηη
РАС .
211
Exercises
. 217
Notes
. 218
Chapter
12.
Hilbertian Fields
. 219
12.1
Hilbert Sets and Reduction Lemmas
. 219
12.2
Hilbert Sets under Separable Algebraic Extensions
. 223
12.3
Purely Inseparable Extensions
. 224
12.4
Imperfect fields
. 228
Exercises
. 229
Notes
. 230
Chapter
13.
The Classical Hilbertian Fields
. 231
13.1
Further Reduction
. 231
13.2
Function Fields over Infinite Fields
. 236
13.3
Global Fields
. 237
13.4
Hilbertian Rings
. 241
13.5
Hilbertianity via Coverings
. 244
x
Table of Contents
13.0
Non-Hilbertian (/-Hilbertian Fields
. 248
13.7
Twisted Wreath Products
. 252
13.8
The Diamond Theorem
. 258
13.9
Weissauers Theorem
. 262
Exercises
. 264
Notes
. 266
Chapter
14.
Nonstandard Structures
. 267
14.1
Higher Order Predicate Calculus
. 267
14.2
Enlargements
. 268
14.3
Concurrent Relations
. 270
14.4
The Existence of Enlargements
. 272
14.5
Examples
. 274
Exercises
. 275
Notes
. 276
Chapter
15.
Nonstandard
Approach
to Hubert's Irreducibility Theorem
. 277
15.1
Criteria for Hilbertianity
. 277
15.2
Arithmetical Primes Versus Functional Primes
. 279
15.3
Fields with the Product Formula
. 281
15.4
Generalized Krull Domains
. 283
15.5
Examples
. 286
Exercises
. 289
Notes
. 290
Chapter
16.
Galois Groups over Hilbertian Fields
. 291
16.1
Galois Groups of Polynomials
. 291
16.2
Stable Polynomials
. 294
16.3
Regular Realization of Finite Abelian Groups
. 298
16.4
Split Embedding Problems with Abelian Kernels
. 302
16.5
Embedding Quadratic Extensions in Z/2raZ-extensions
. . . 306
16.6
Zp-Extensions of Hilbertian Fields
. 308
16.7
Symmetric and Alternating Groups over Hilbertian Fields
. 315
16.8
GAR-Realizations
. 321
16.9
Embedding Problems over Hilbertian Fields
. 325
16.10
Finitely Generated
Profinite
Groups
. 328
16.11
Abelian Extensions of Hilbertian Fields
. 332
16.12
Regularity of Finite Groups
over Complete Discrete Valued Fields
. 334
Exercises
. 335
Notes
. 336
Chapter
17.
Free Profinite Groups
. 338
17.1
The Rank of a Profinite Group
. 338
Table
of Contents
xi
17.2
Profinite
Completions of Groups
. 340
17.3
Formations of Finite Groups
. 344
17.4
Free pro
-С
Groups
. 346
17.5
Subgroups of Free Discrete Groups
. 350
17.6
Open Subgroups of Free Profinite Groups
. 358
17.7
An Embedding Property
. 360
Exercises
. 361
Notes
. 362
Chapter
18.
The
Haar
Measure
. 363
18.1
The
Haar
Measure of a Profinite Group
. 363
18.2
Existence of the
Haar
Measure
. 366
18.3
Independence.
. 370
18.4
Cartesian Product of
Haar
Measures
. 376
18.5
The
Haar
Measure of the Absolute Galois Group
. 378
18.6
The
PAC
Nullstellensatz . 380
18.7
The Bottom Theorem
. 382
18.8
PAC
Fields over Uncountable Hilbertian Fields
. 386
18.9
On the Stability of Fields
. 390
18.10
PAC
Galois Extensions of Hilbertian Fields
. 394
18.11
Algebraic Groups
. 397
Exercises
. 400
Notes
. 401
Chapter
19.
Effective Field Theory and Algebraic Geometry
. . . 403
19.1
Presented Rings and Fields
. 403
19.2
Extensions of Presented Fields
. 406
19.3
Galois Extensions of Presented Fields
. 411
19.4
The Algebraic and Separable Closures of Presented Fields
. 412
19.5
Constructive Algebraic Geometry
. 413
19.6
Presented Rings and
Constructible
Sets
. 422
19.7
Basic Normal Stratification
. 425
Exercises
. 427
Notes
. 428
Chapter
20.
The Elementary Theory of
е
-Free
PAC
Fields
. 429
20.1
Kj-Saturated
PAC
Fields
. 429
20.2
The Elementary Equivalence Theorem
of Hi-Saturated
PAC
Fields
. 430
20.3
Elementary Equivalence of
PAC
Fields
. 433
20.4
On
е
-Free
PAC
Fields
. 436
20.5
The Elementary Theory of Perfect
е
-Free
PAC
Fields
. 438
20.6
The Probable Truth of a Sentence
. 440
20.7
Change of Base Field
. 442
20.8
The Fields Ks(au.,ae)
. 444
xii
Table
of
Contents
20.9
The Transfer Theorem
. 440
20.10
The Elementary Theory of Finite Fields
. 448
Exercises
. 451
Notes
. 453
Chapter
21.
Problems of Arithmetical Geometry
. 454
21.1
The Decomposition-Intersection Procedure
. 454
21.2
Ci-Fields and Weakly Q-Fields
. 455
21.3
Perfect
PAC
Fields which are
d
. 460
21.4
The Existential Theory of
PAC
Fields
. 462
21.5 Kronecker
Classes of Number Fields
. 463
21.6
Davenport's Problem
. 467
21.7
On permutation Groups
. 472
21.8 Schurs
Conjecture
. 479
21.9
Generalized Carlitz's Conjecture
. 489
Exercises
. 493
Notes
. 495
Chapter
22.
Projective
Groups and Frattini Covers
. 497
22.1
The Frattini Groups of
a Profinite
Group
. 497
22.2
Cartesian Squares
. 499
22.3
On C-Projective Groups
. 502
22.4
Projective
Groups
. 506
22.5
Frattini Covers
. 508
22.6
The Universal Frattini Cover
. 513
22.7
Projective
Pro-p-Groups
. 515
22.8
Supernatural Numbers
. 520
22.9
The Sylow Theorems
. 522
22.10
On Complements of Normal Subgroups
. 524
22.11
The Universal Frattini p-Cover
. 528
22.12
Examples of Universal Frattini p-Covers
. 532
22.13
The Special Linear Group SL(2,ZP)
. 534
22.14
The General Linear Group GL(2,Zp)
. 537
Exercises
. 539
Notes
. 542
Chapter
23.
PAC
Fields and
Projective
Absolute Galois Groups
. . 544
23.1
Projective
Groups as Absolute Galois Groups
. 544
23.2
Countably Generated
Projective
Groups
. 546
23.3
Perfect
PAC
Fields of Bounded Corank
. 549
23.4
Basic Elementary Statements
. 550
23.5
Reduction Steps
. 554
23.6
Application of Ultraproducts
. 558
Exercises
. 561
Notes
. 561
Table
of Contents
xiii
Chapter
24.
Frobenius Fields
. 562
24.1
The Field Crossing Argument
. 562
24.2
The
Beckmann-Black
Problem
. 565
24.3
The Embedding Property and Maximal Frattini Covers
. . 567
24.4
The Smallest Embedding Cover of
a Prorinite
Group
. 569
24.5
A Decision Procedure
. 574
24.6
Examples
. 576
24.7
Non-pro
j
ective Smallest Embedding Cover
. 579
24.8
A Theorem of Iwasawa
. 581
24.9
Free
Profinite
Groups of at most Countable Rank
. 583
24.10
Application of the
Nielsen-Schreier
Formula
. 586
Exercises
. 591
Notes
. 592
Chapter
25.
Free Profinite Groups of Infinite Rank
. 594
25.1
Characterization of Free Profinite Groups
by Embedding Problems
. 595
25.2
Applications of Theorem
25.1.7 . 601
25.3
The Pro
-С
Completion of a Free Discrete Group
. 604
25.4
The Group Theoretic Diamond Theorem
. 606
25.5
The Melnikov Group of a Profinite Group
. 613
25.6
Homogeneous Pro
-С
Groups
. 615
25.7
The S-rank of Closed Normal Subgroups
. 620
25.8
Closed Normal Subgroups with a Basis Element
. 623
25.9
Accessible Subgroups
. 625
Notes
. 633
Chapter
26.
Random Elements in Free Profinite Groups
. 635
26.1
Random Elements in a Free Profinite Group
. 635
26.2
Random Elements in Free pio-p Groups
. 640
26.3
Random
е
-tuples in
Żn
. 642
26.4
On the Index of Normal Subgroups
Generated by Random Elements
. 646
26.5
Freeness of Normal Subgroups
Generated by Random Elements
. 651
Notes
. 654
Chapter
27.
Omega-Free
PAC
Fields
. 655
27.1
Model Companions
. 655
27.2
The Model Companion in an Augmented Theory of Fields
. 659
27.3
New Non-Classical Hilbertian Fields
. 664
27.4
An abundance of
ω
-Free
PAC
Fields
. 667
Notes
. 670
xiv
Table
of
Contents
Chapter
28.
Uiidecidability
. 671
28.1
Turing
Machines
. 671
28.2
Computation of Functions by Turing Machines
. 672
28.3
Recursive Inseparability of Sets of Turing Machines
. 676
28.4
The Predicate Calculus
. 679
28.5
Undecidability in the Theory of Graphs
. 682
28.6
Assigning Graphs to
Profinite
Groups
. 687
28.7
The Graph Conditions
. 688
28.8
Assigning
Profinite
Groups to Graphs
. 690
28.9
Assigning Fields to Graphs
. 694
28.10
Interpretation of the Theory of Graphs
in the Theory of Fields
. 694
Exercises
. 697
Notes
. 697
Chapter
29.
Algebraically Closed Fields with
Distinguished Automorphisms
. . 698
29.1
The Base Field
К
. 698
29.2
Coding in
РАС
Fields with Monadic Quantifiers
. 700
29.3
The Theory of Almost all
{Ќ,аи.,ае)'ѕ
. 704
29.4
The Probability of Truth Sentences
. 706
Chapter
30.
Galois Stratification
. 708
30.1
The Artin Symbol
. 708
30.2
Conjugacy Domains under Projection
. 710
30.3
Normal Stratification
. 715
30.4
Elimination of One Variable
. 717
30.5
The Complete Elimination Procedure
. 720
30.6
Model-Theoretic Applications
. 722
30.7
A Limit of Theories
. 725
Exercises
. 726
Notes
. 729
Chapter
31.
Galois Stratification over Finite Fields
. 730
31.1
The Elementary Theory of Frobenius Fields
. 730
31.2
The Elementary Theory of Finite Fields
. 735
31.3
Near Rationality of the
Zeta
Function of a Galois Formula
. 739
Exercises
. 748
Notes
. 750
Chapter
32.
Problems of Field Arithmetic
. 751
32.1
Open Problems of the First Edition
. 751
32.2
Open Problems of the Second Edition
. 754
32.3
Open problems
. 758
Table
of
Contents
xv
References
.761
Index
.780 |
| adam_txt |
Table
of
Contents
Chapter
1.
Infinite Galois Theory and Profmite Groups
.1
1.1
Inverse Limits
.1
1.2
Profinite
Groups
.4
1.3
Infinite Galois Theory
.9
1.4
The p-adic Integers and the
Prüfer
Group
. 12
1.5
The Absolute Galois Group of a Finite Field
. 15
Exercises
. 16
Notes
. 18
Chapter
2.
Valuations and Linear Disjointness
. 19
2.1
Valuations, Places, and Valuation Rings
. 19
2.2
Discrete Valuations
. 21
2.3
Extensions of Valuations and Places
. 24
2.4
Integral Extensions and Dedekind Domains
. 30
2.5
Linear Disjointness of Fields
. 34
2.6
Separable, Regular, and Primary Extensions
. 38
2.7
The Imperfect Degree of a Field
. 44
2.8
Derivatives
. 48
Exercises
. 50
Notes
. 51
Chapter
3.
Algebraic Function Fields of One Variable
. 52
3.1
Function Fields of One Variable
. 52
3.2
The Riemann-Roch Theorem
. 54
3.3
Holomorphy Rings
. 56
3.4
Extensions of Function Fields
. 59
3.5
Completions
. 61
3.6
The Different
. 67
3.7
Hyperelliptic Fields
. 70
3.8
Hyperelliptic Fields with a Rational quadratic Subfield
. 73
Exercises
. 75
Notes
. 76
Chapter
4.
The Riemann Hypothesis for Function Fields
. 77
4.1
Class Numbers
. 77
4.2
Zeta
Functions
. 79
4.3
Zeta
Functions under Constant Field Extensions
. 81
4.4
The Functional Equation
. 82
4.5
The Riemann Hypothesis and Degree
1
Prime Divisors
. 84
4.6
Reduction Steps
. 86
4.7
An Upper Bound
. 87
4.8
A Lower Bound
. 89
viii Table of
Contents
Exercises .
91
Notes
. 93
Chapter
5.
Plane
Curves .
95
5.1
Affine
and
Projective
Plane Curves
. 95
5.2
Points and prime divisors
. 97
5.3
The Genus of a Plane Curve
. 99
5.4
Points on a Curve over a Finite Field
. 104
Exercises
. 105
Notes
. 106
Chapter
6.
The Chebotarev Density Theorem
. 107
6.1
Decomposition Groups
. 107
6.2
The Artin Symbol over Global Fields
.
Ill
6.3
Dirichlet Density
. 113
6.4
Function Fields
. 115
6.5
Number Fields
. 121
Exercises
. 129
Notes
. 130
Chapter
7.
Ultraproducts
. 132
7.1
First Order Predicate Calculus
. 132
7.2
Structures
. 134
7.3
Models
. 135
7.4
Elementary Substructures
. 137
7.5
Ultrafilters
. 138
7.6
Regular
Ultrafilters
. 139
7.7
Ultraproducts
. 141
7.8
Regular Ultraproducts
. 145
7.9
Xonprincipal Ultraproducts of Finite Fields
. 147
Exercises
. 147
Notes
. 148
Chapter
8.
Decision Procedures
. 149
8.1
Deduction Theory
. 149
8.2
Gödel's
Completeness Theorem
. 152
8.3
Primitive Recursive Functions
. 154
8.4
Primitive Recursive Relations
. 156
8.5
Recursive Functions
. 157
8.6
Recursive and Primitive Recursive Procedures
. 159
8.7
A Reduction Step in Decidability Procedures
. 160
Exercises
. 161
Notes
. 162
Table
of Contents
ix
Chapter
9.
Algebraically Closed Fields
. 163
9.1
Elimination of Quantifiers
. 163
9.2
A Quantifiers Elimination Procedure
. 165
9.3
Effectiveness
. 168
9.4
Applications
. . 169
Exercises
. 170
Notes
. 170
Chapter
10.
Elements of Algebraic Geometry
. 172
10.1
Algebraic Sets
. 172
10.2
Varieties
. 175
10.3
Substitutions in Irreducible Polynomials
. 176
10.4
Rational Maps
. 178
10.5 Hyperplane
Sections
. 180
10.6
Descent
. 182
10.7
Projective
Varieties
. 185
10.8
About the Language of Algebraic Geometry
. 187
Exercises
. 190
Notes
. 191
Chapter
11.
Pseudo
Algebraically Closed Fields
. 192
11.1
PAC
Fields
.". 192
11.2
Reduction to Plane Curves
. 193
11.3
The
PAC
Property is an Elementary Statement
. 199
11.4
PAC
Fields of Positive Characteristic
. 201
11.5
PAC
Fields with Valuations
. 203
11.6
The Absolute Galois Group of
a PAC
Field
. 207
11.7
A non-PAC
Field
К
with
ϋΓίηη
РАС .
211
Exercises
. 217
Notes
. 218
Chapter
12.
Hilbertian Fields
. 219
12.1
Hilbert Sets and Reduction Lemmas
. 219
12.2
Hilbert Sets under Separable Algebraic Extensions
. 223
12.3
Purely Inseparable Extensions
. 224
12.4
Imperfect fields
. 228
Exercises
. 229
Notes
. 230
Chapter
13.
The Classical Hilbertian Fields
. 231
13.1
Further Reduction
. 231
13.2
Function Fields over Infinite Fields
. 236
13.3
Global Fields
. 237
13.4
Hilbertian Rings
. 241
13.5
Hilbertianity via Coverings
. 244
x
Table of Contents
13.0
Non-Hilbertian (/-Hilbertian Fields
. 248
13.7
Twisted Wreath Products
. 252
13.8
The Diamond Theorem
. 258
13.9
Weissauers Theorem
. 262
Exercises
. 264
Notes
. 266
Chapter
14.
Nonstandard Structures
. 267
14.1
Higher Order Predicate Calculus
. 267
14.2
Enlargements
. 268
14.3
Concurrent Relations
. 270
14.4
The Existence of Enlargements
. 272
14.5
Examples
. 274
Exercises
. 275
Notes
. 276
Chapter
15.
Nonstandard
Approach
to Hubert's Irreducibility Theorem
. 277
15.1
Criteria for Hilbertianity
. 277
15.2
Arithmetical Primes Versus Functional Primes
. 279
15.3
Fields with the Product Formula
. 281
15.4
Generalized Krull Domains
. 283
15.5
Examples
. 286
Exercises
. 289
Notes
. 290
Chapter
16.
Galois Groups over Hilbertian Fields
. 291
16.1
Galois Groups of Polynomials
. 291
16.2
Stable Polynomials
. 294
16.3
Regular Realization of Finite Abelian Groups
. 298
16.4
Split Embedding Problems with Abelian Kernels
. 302
16.5
Embedding Quadratic Extensions in Z/2raZ-extensions
. . . 306
16.6
Zp-Extensions of Hilbertian Fields
. 308
16.7
Symmetric and Alternating Groups over Hilbertian Fields
. 315
16.8
GAR-Realizations
. 321
16.9
Embedding Problems over Hilbertian Fields
. 325
16.10
Finitely Generated
Profinite
Groups
. 328
16.11
Abelian Extensions of Hilbertian Fields
. 332
16.12
Regularity of Finite Groups
over Complete Discrete Valued Fields
. 334
Exercises
. 335
Notes
. 336
Chapter
17.
Free Profinite Groups
. 338
17.1
The Rank of a Profinite Group
. 338
Table
of Contents
xi
17.2
Profinite
Completions of Groups
. 340
17.3
Formations of Finite Groups
. 344
17.4
Free pro
-С
Groups
. 346
17.5
Subgroups of Free Discrete Groups
. 350
17.6
Open Subgroups of Free Profinite Groups
. 358
17.7
An Embedding Property
. 360
Exercises
. 361
Notes
. 362
Chapter
18.
The
Haar
Measure
. 363
18.1
The
Haar
Measure of a Profinite Group
. 363
18.2
Existence of the
Haar
Measure
. 366
18.3
Independence.
. 370
18.4
Cartesian Product of
Haar
Measures
. 376
18.5
The
Haar
Measure of the Absolute Galois Group
. 378
18.6
The
PAC
Nullstellensatz . 380
18.7
The Bottom Theorem
. 382
18.8
PAC
Fields over Uncountable Hilbertian Fields
. 386
18.9
On the Stability of Fields
. 390
18.10
PAC
Galois Extensions of Hilbertian Fields
. 394
18.11
Algebraic Groups
. 397
Exercises
. 400
Notes
. 401
Chapter
19.
Effective Field Theory and Algebraic Geometry
. . . 403
19.1
Presented Rings and Fields
. 403
19.2
Extensions of Presented Fields
. 406
19.3
Galois Extensions of Presented Fields
. 411
19.4
The Algebraic and Separable Closures of Presented Fields
. 412
19.5
Constructive Algebraic Geometry
. 413
19.6
Presented Rings and
Constructible
Sets
. 422
19.7
Basic Normal Stratification
. 425
Exercises
. 427
Notes
. 428
Chapter
20.
The Elementary Theory of
е
-Free
PAC
Fields
. 429
20.1
Kj-Saturated
PAC
Fields
. 429
20.2
The Elementary Equivalence Theorem
of Hi-Saturated
PAC
Fields
. 430
20.3
Elementary Equivalence of
PAC
Fields
. 433
20.4
On
е
-Free
PAC
Fields
. 436
20.5
The Elementary Theory of Perfect
е
-Free
PAC
Fields
. 438
20.6
The Probable Truth of a Sentence
. 440
20.7
Change of Base Field
. 442
20.8
The Fields Ks(au.,ae)
. 444
xii
Table
of
Contents
20.9
The Transfer Theorem
. 440
20.10
The Elementary Theory of Finite Fields
. 448
Exercises
. 451
Notes
. 453
Chapter
21.
Problems of Arithmetical Geometry
. 454
21.1
The Decomposition-Intersection Procedure
. 454
21.2
Ci-Fields and Weakly Q-Fields
. 455
21.3
Perfect
PAC
Fields which are
d
. 460
21.4
The Existential Theory of
PAC
Fields
. 462
21.5 Kronecker
Classes of Number Fields
. 463
21.6
Davenport's Problem
. 467
21.7
On permutation Groups
. 472
21.8 Schurs
Conjecture
. 479
21.9
Generalized Carlitz's Conjecture
. 489
Exercises
. 493
Notes
. 495
Chapter
22.
Projective
Groups and Frattini Covers
. 497
22.1
The Frattini Groups of
a Profinite
Group
. 497
22.2
Cartesian Squares
. 499
22.3
On C-Projective Groups
. 502
22.4
Projective
Groups
. 506
22.5
Frattini Covers
. 508
22.6
The Universal Frattini Cover
. 513
22.7
Projective
Pro-p-Groups
. 515
22.8
Supernatural Numbers
. 520
22.9
The Sylow Theorems
. 522
22.10
On Complements of Normal Subgroups
. 524
22.11
The Universal Frattini p-Cover
. 528
22.12
Examples of Universal Frattini p-Covers
. 532
22.13
The Special Linear Group SL(2,ZP)
. 534
22.14
The General Linear Group GL(2,Zp)
. 537
Exercises
. 539
Notes
. 542
Chapter
23.
PAC
Fields and
Projective
Absolute Galois Groups
. . 544
23.1
Projective
Groups as Absolute Galois Groups
. 544
23.2
Countably Generated
Projective
Groups
. 546
23.3
Perfect
PAC
Fields of Bounded Corank
. 549
23.4
Basic Elementary Statements
. 550
23.5
Reduction Steps
. 554
23.6
Application of Ultraproducts
. 558
Exercises
. 561
Notes
. 561
Table
of Contents
xiii
Chapter
24.
Frobenius Fields
. 562
24.1
The Field Crossing Argument
. 562
24.2
The
Beckmann-Black
Problem
. 565
24.3
The Embedding Property and Maximal Frattini Covers
. . 567
24.4
The Smallest Embedding Cover of
a Prorinite
Group
. 569
24.5
A Decision Procedure
. 574
24.6
Examples
. 576
24.7
Non-pro
j
ective Smallest Embedding Cover
. 579
24.8
A Theorem of Iwasawa
. 581
24.9
Free
Profinite
Groups of at most Countable Rank
. 583
24.10
Application of the
Nielsen-Schreier
Formula
. 586
Exercises
. 591
Notes
. 592
Chapter
25.
Free Profinite Groups of Infinite Rank
. 594
25.1
Characterization of Free Profinite Groups
by Embedding Problems
. 595
25.2
Applications of Theorem
25.1.7 . 601
25.3
The Pro
-С
Completion of a Free Discrete Group
. 604
25.4
The Group Theoretic Diamond Theorem
. 606
25.5
The Melnikov Group of a Profinite Group
. 613
25.6
Homogeneous Pro
-С
Groups
. 615
25.7
The S-rank of Closed Normal Subgroups
. 620
25.8
Closed Normal Subgroups with a Basis Element
. 623
25.9
Accessible Subgroups
. 625
Notes
. 633
Chapter
26.
Random Elements in Free Profinite Groups
. 635
26.1
Random Elements in a Free Profinite Group
. 635
26.2
Random Elements in Free pio-p Groups
. 640
26.3
Random
е
-tuples in
Żn
. 642
26.4
On the Index of Normal Subgroups
Generated by Random Elements
. 646
26.5
Freeness of Normal Subgroups
Generated by Random Elements
. 651
Notes
. 654
Chapter
27.
Omega-Free
PAC
Fields
. 655
27.1
Model Companions
. 655
27.2
The Model Companion in an Augmented Theory of Fields
. 659
27.3
New Non-Classical Hilbertian Fields
. 664
27.4
An abundance of
ω
-Free
PAC
Fields
. 667
Notes
. 670
xiv
Table
of
Contents
Chapter
28.
Uiidecidability
. 671
28.1
Turing
Machines
. 671
28.2
Computation of Functions by Turing Machines
. 672
28.3
Recursive Inseparability of Sets of Turing Machines
. 676
28.4
The Predicate Calculus
. 679
28.5
Undecidability in the Theory of Graphs
. 682
28.6
Assigning Graphs to
Profinite
Groups
. 687
28.7
The Graph Conditions
. 688
28.8
Assigning
Profinite
Groups to Graphs
. 690
28.9
Assigning Fields to Graphs
. 694
28.10
Interpretation of the Theory of Graphs
in the Theory of Fields
. 694
Exercises
. 697
Notes
. 697
Chapter
29.
Algebraically Closed Fields with
Distinguished Automorphisms
. . 698
29.1
The Base Field
К
. 698
29.2
Coding in
РАС
Fields with Monadic Quantifiers
. 700
29.3
The Theory of Almost all
{Ќ,аи.,ае)'ѕ
. 704
29.4
The Probability of Truth Sentences
. 706
Chapter
30.
Galois Stratification
. 708
30.1
The Artin Symbol
. 708
30.2
Conjugacy Domains under Projection
. 710
30.3
Normal Stratification
. 715
30.4
Elimination of One Variable
. 717
30.5
The Complete Elimination Procedure
. 720
30.6
Model-Theoretic Applications
. 722
30.7
A Limit of Theories
. 725
Exercises
. 726
Notes
. 729
Chapter
31.
Galois Stratification over Finite Fields
. 730
31.1
The Elementary Theory of Frobenius Fields
. 730
31.2
The Elementary Theory of Finite Fields
. 735
31.3
Near Rationality of the
Zeta
Function of a Galois Formula
. 739
Exercises
. 748
Notes
. 750
Chapter
32.
Problems of Field Arithmetic
. 751
32.1
Open Problems of the First Edition
. 751
32.2
Open Problems of the Second Edition
. 754
32.3
Open problems
. 758
Table
of
Contents
xv
References
.761
Index
.780 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Fried, Michael D. 1942- Yardēn, Moše 1942- |
| author_GND | (DE-588)172081203 (DE-588)122073932 |
| author_facet | Fried, Michael D. 1942- Yardēn, Moše 1942- |
| author_role | aut aut |
| author_sort | Fried, Michael D. 1942- |
| author_variant | m d f md mdf m y my |
| building | Verbundindex |
| bvnumber | BV023355978 |
| classification_rvk | SK 130 SK 180 SK 230 |
| ctrlnum | (OCoLC)244290764 (DE-599)DNB988747138 |
| 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 13 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | Third edition, revised by Moshe Jarden |
| format | Book |
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| id | DE-604.BV023355978 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:06:58Z |
| indexdate | 2024-07-20T09:42:41Z |
| institution | BVB |
| isbn | 9783540772699 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016539518 |
| oclc_num | 244290764 |
| open_access_boolean | |
| owner | DE-29T DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-703 DE-188 |
| owner_facet | DE-29T DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-703 DE-188 |
| physical | xxiii, 792 Seiten Illustrationen 235 mm x 155 mm, 1310 gr. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Ergebnisse der Mathematik und ihrer Grenzgebiete |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |
| spelling | Fried, Michael D. 1942- Verfasser (DE-588)172081203 aut Field arithmetic Michael D. Fried, Moshe Jarden Third edition, revised by Moshe Jarden Berlin Springer [2008] © 2008 xxiii, 792 Seiten Illustrationen 235 mm x 155 mm, 1310 gr. txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge volume 11 Algebraischer Körper (DE-588)4141852-9 gnd rswk-swf Absoluter Klassenkörper (DE-588)4132442-0 gnd rswk-swf Algebraischer Zahlkörper (DE-588)4068537-8 gnd rswk-swf Pseudoalgebraisch abgeschlossener Körper (DE-588)4132443-2 gnd rswk-swf Proendliche Gruppe (DE-588)4132444-4 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Funktionenkörper (DE-588)4155688-4 gnd rswk-swf Algebraischer Funktionenkörper (DE-588)4141850-5 gnd rswk-swf Algebraischer Zahlkörper (DE-588)4068537-8 s Pseudoalgebraisch abgeschlossener Körper (DE-588)4132443-2 s DE-604 Absoluter Klassenkörper (DE-588)4132442-0 s Proendliche Gruppe (DE-588)4132444-4 s 1\p DE-604 Algebraische Zahlentheorie (DE-588)4001170-7 s 2\p DE-604 Algebraischer Funktionenkörper (DE-588)4141850-5 s 3\p DE-604 Funktionenkörper (DE-588)4155688-4 s 4\p DE-604 Algebraischer Körper (DE-588)4141852-9 s 5\p DE-604 Yardēn, Moše 1942- Verfasser (DE-588)122073932 aut Nachgedruckt als 978-3-642-09594-8 Erscheint auch als Online-Ausgabe 978-3-540-77270-5 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge ; volume 11 (DE-604)BV000899194 11 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3109440&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016539518&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 5\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Fried, Michael D. 1942- Yardēn, Moše 1942- Field arithmetic Ergebnisse der Mathematik und ihrer Grenzgebiete Algebraischer Körper (DE-588)4141852-9 gnd Absoluter Klassenkörper (DE-588)4132442-0 gnd Algebraischer Zahlkörper (DE-588)4068537-8 gnd Pseudoalgebraisch abgeschlossener Körper (DE-588)4132443-2 gnd Proendliche Gruppe (DE-588)4132444-4 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Funktionenkörper (DE-588)4155688-4 gnd Algebraischer Funktionenkörper (DE-588)4141850-5 gnd |
| subject_GND | (DE-588)4141852-9 (DE-588)4132442-0 (DE-588)4068537-8 (DE-588)4132443-2 (DE-588)4132444-4 (DE-588)4001170-7 (DE-588)4155688-4 (DE-588)4141850-5 |
| title | Field arithmetic |
| title_auth | Field arithmetic |
| title_exact_search | Field arithmetic |
| title_exact_search_txtP | Field arithmetic |
| title_full | Field arithmetic Michael D. Fried, Moshe Jarden |
| title_fullStr | Field arithmetic Michael D. Fried, Moshe Jarden |
| title_full_unstemmed | Field arithmetic Michael D. Fried, Moshe Jarden |
| title_short | Field arithmetic |
| title_sort | field arithmetic |
| topic | Algebraischer Körper (DE-588)4141852-9 gnd Absoluter Klassenkörper (DE-588)4132442-0 gnd Algebraischer Zahlkörper (DE-588)4068537-8 gnd Pseudoalgebraisch abgeschlossener Körper (DE-588)4132443-2 gnd Proendliche Gruppe (DE-588)4132444-4 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Funktionenkörper (DE-588)4155688-4 gnd Algebraischer Funktionenkörper (DE-588)4141850-5 gnd |
| topic_facet | Algebraischer Körper Absoluter Klassenkörper Algebraischer Zahlkörper Pseudoalgebraisch abgeschlossener Körper Proendliche Gruppe Algebraische Zahlentheorie Funktionenkörper Algebraischer Funktionenkörper |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3109440&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=016539518&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000899194 |
| work_keys_str_mv | AT friedmichaeld fieldarithmetic AT yardenmose fieldarithmetic |