Field arithmetic:
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| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin
Springer
[2008]
|
| Ausgabe: | Third edition, revised by Moshe Jarden |
| Schriftenreihe: | Ergebnisse der Mathematik und ihrer Grenzgebiete
3. Folge ; volume 11 |
| Schlagwörter: | |
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| Umfang: | xxiii, 792 Seiten Illustrationen 235 mm x 155 mm, 1310 gr. |
| 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_ | 1819381935605022720 |
|---|---|
| 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
|
| any_adam_object | 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 |
| edition | Third edition, revised by Moshe Jarden |
| format | Book |
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| id | DE-604.BV023355978 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:14:22Z |
| 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 |
| spellingShingle | Fried, Michael D. 1942- Yardēn, Moše 1942- Field arithmetic Ergebnisse der Mathematik und ihrer Grenzgebiete Pseudoalgebraisch abgeschlossener Körper (DE-588)4132443-2 gnd Algebraischer Körper (DE-588)4141852-9 gnd Absoluter Klassenkörper (DE-588)4132442-0 gnd Proendliche Gruppe (DE-588)4132444-4 gnd Funktionenkörper (DE-588)4155688-4 gnd Algebraischer Funktionenkörper (DE-588)4141850-5 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Algebraischer Zahlkörper (DE-588)4068537-8 gnd |
| subject_GND | (DE-588)4132443-2 (DE-588)4141852-9 (DE-588)4132442-0 (DE-588)4132444-4 (DE-588)4155688-4 (DE-588)4141850-5 (DE-588)4001170-7 (DE-588)4068537-8 |
| title | Field arithmetic |
| title_auth | Field arithmetic |
| title_exact_search | 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 | Pseudoalgebraisch abgeschlossener Körper (DE-588)4132443-2 gnd Algebraischer Körper (DE-588)4141852-9 gnd Absoluter Klassenkörper (DE-588)4132442-0 gnd Proendliche Gruppe (DE-588)4132444-4 gnd Funktionenkörper (DE-588)4155688-4 gnd Algebraischer Funktionenkörper (DE-588)4141850-5 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Algebraischer Zahlkörper (DE-588)4068537-8 gnd |
| topic_facet | Pseudoalgebraisch abgeschlossener Körper Algebraischer Körper Absoluter Klassenkörper Proendliche Gruppe Funktionenkörper Algebraischer Funktionenkörper Algebraische Zahlentheorie Algebraischer Zahlkörper |
| 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 |