Field Arithmetic

by Fried, Michael D.; Jarden, Moshe

• ISBN13: 9783540772699
• ISBN10: 3540772693
• Edition: 3rd
• Format: Hardcover
• Copyright: 2008-07-04
• Publisher: Springer Verlag

Summary

Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements. Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)? The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.

Author Biography

Moshe Jarden (revised and considerably enlarged the book in 2004 (2nd edition) and revised again in 2007 (the present 3rd edition).  Born on 23 August, 1942 in Tel Aviv, Israel.Education:Ph.D. 1969 at the Hebrew University of Jerusalem on"Rational Points of Algebraic Varieties over Large Algebraic Fields".Thesis advisor: H. Furstenberg.Habilitation at Heidelberg University, 1972, on"Model Theory Methods in the Theory of Fields".Positions:Dozent, Heidelberg University, 1973-1974.Seniour Lecturer, Tel Aviv University, 1974-1978Associate Professor, Tel Aviv University, 1978-1982Professor, Tel Aviv University, 1982-Incumbent of the Cissie and Aaron Beare Chair,Tel Aviv University. 1998-Academic and Professional AwardsFellowship of Alexander von Humboldt-Stiftung in Heidelberg, 1971-1973.Fellowship of Minerva Foundation, 1982.Chairman of the Israel Mathematical Society, 1986-1988.Member of the Institute for Advanced Study, Princeton, 1983, 1988.Editor of the Israel Journal of Mathematics, 1992-.Landau Prize for the book "Field Arithmetic". 1987.Director of the Minkowski Center for Geometry founded by theMinerva Foundation, 1997-.L. Meitner-A.v.Humboldt Research Prize, 2001Member, Max-Planck Institut f\"ur Mathematik in Bonn, 2001. 

Table of Contents

Infinite Galois Theory and Profinite Groups	p. 1
Inverse Limits	p. 1
Profinite Groups	p. 4
Infinite Galois Theory	p. 9
The p-adic Integers and the Prüfer Group	p. 12
The Absolute Galois Group of a Finite Field	p. 15
Exercises	p. 16
Notes	p. 18
Valuations and Linear Disjointness	p. 19
Valuations, Places, and Valuation Rings	p. 19
Discrete Valuations	p. 21
Extensions of Valuations and Places	p. 24
Integral Extensions and Dedekind Domains	p. 30
Linear Disjointness of Fields	p. 34
Separable, Regular, and Primary Extensions	p. 38
The Imperfect Degree of a Field	p. 44
Derivatives	p. 48
Exercises	p. 50
Notes	p. 51
Algebraic Function Fields of One Variable	p. 52
Function Fields of One Variable	p. 52
The Riemann-Roch Theorem	p. 54
Holomorphy Rings	p. 56
Extensions of Function Fields	p. 59
Completions	p. 61
The Different	p. 67
Hyperelliptic Fields	p. 70
Hyperelliptic Fields with a Rational quadratic Subfield	p. 73
Exercises	p. 75
Notes	p. 76
The Riemann Hypothesis for Function Fields	p. 77
Class Numbers	p. 77
Zeta Functions	p. 79
Zeta Functions under Constant Field Extensions	p. 81
The Functional Equation	p. 82
The Riemann Hypothesis and Degree 1 Prime Divisors	p. 84
Reduction Steps	p. 86
An Upper Bound	p. 87
A Lower Bound	p. 89
Exercises	p. 91
Notes	p. 93
Plane Curves	p. 95
Affine and Projective Plane Curves	p. 95
Points and prime divisors	p. 97
The Genus of a Plane Curve	p. 99
Points on a Curve over a Finite Field	p. 104
Exercises	p. 105
Notes	p. 106
The Chebotarev Density Theorem	p. 107
Decomposition Groups	p. 107
The Artin Symbol over Global Fields	p. 111
Dirichlet Density	p. 113
Function Fields	p. 115
Number Fields	p. 121
Exercises	p. 129
Notes	p. 130
Ultraproducts	p. 132
First Order Predicate Calculus	p. 132
Structures	p. 134
Models	p. 135
Elementary Substructures	p. 137
Ultrafilters	p. 138
Regular Ultrafilters	p. 139
Ultraproducts	p. 141
Regular Ultraproducts	p. 145
Nonprincipal Ultraproducts of Finite Fields	p. 147
Exercises	p. 147
Notes	p. 148
Decision Procedures	p. 149
Deduction Theory	p. 149
Gödel's Completeness Theorem	p. 152
Primitive Recursive Functions	p. 154
Primitive Recursive Relations	p. 156
Recursive Functions	p. 157
Recursive and Primitive Recursive Procedures	p. 159
A Reduction Step in Decidability Procedures	p. 160
Exercises	p. 161
Notes	p. 162
Algebraically Closed Fields	p. 163
Elimination of Quantifiers	p. 163
A Quantifiers Elimination Procedure	p. 165
Effectiveness	p. 168
Applications	p. 169
Exercises	p. 170
Notes	p. 170
Elements of Algebraic Geometry	p. 172
Algebraic Sets	p. 172
Varieties	p. 175
Substitutions in Irreducible Polynomials	p. 176
Rational Maps	p. 178
Hyperplane Sections	p. 180
Descent	p. 182
Projective Varieties	p. 185
About the Language of Algebraic Geometry	p. 187
Exercises	p. 190
Notes	p. 191
Pseudo Algebraically Closed Fields	p. 192
PAC Fields	p. 192
Reduction to Plane Curves	p. 193
The PAC Property is an Elementary Statement	p. 199
PAC Fields of Positive Characteristic	p. 201
PAC Fields with Valuations	p. 203
The Absolute Galois Group of a PAC Field	p. 207
A non-PAC Field K with Kins PAC	p. 211
Exercises	p. 217
Notes	p. 218
Hilbertian Fields	p. 219
Hilbert Sets and Reduction Lemmas	p. 219
Hilbert Sets under Separable Algebraic Extensions	p. 223
Purely Inseparable Extensions	p. 224
Imperfect fields	p. 228
Exercises	p. 229
Notes	p. 230
The Classical Hilbertian Fields	p. 231
Further Reduction	p. 231
Function Fields over Infinite Fields	p. 236
Global Fields	p. 237
Hilbertian Rings	p. 241
Hilbertianity via Coverings	p. 244
Non-Hilbertian g-Hilbertian Fields	p. 248
Twisted Wreath Products	p. 252
The Diamond Theorem	p. 258
Weissauer's Theorem	p. 262
Exercises	p. 264
Notes	p. 266
Nonstandard Structures	p. 267
Higher Order Predicate Calculus	p. 267
Enlargements	p. 268
Concurrent Relations	p. 270
The Existence of Enlargements	p. 272
Examples	p. 274
Exercises	p. 275
Notes	p. 276
Nonstandard Approach to Hilbert's Irreducibility Theorem	p. 277
Criteria for Hilbertianity	p. 277
Arithmetical Primes Versus Functional Primes	p. 279
Fields with the Product Formula	p. 281
Generalized Krull Domains	p. 283
Examples	p. 286
Exercises	p. 289
Notes	p. 290
Galois Groups over Hilbertian Fields	p. 291
Galois Groups of Polynomials	p. 291
Stable Polynomials	p. 294
Regular Realization of Finite Abelian Groups	p. 298
Split Embedding Problems with Abelian Kernels	p. 302
Embedding Quadratic Extensions in <$>{\cal Z}/2^n{\cal Z}<$>-extensions	p. 306
<$>{\cal Z}_p<$>-Extensions of Hilbertian Fields	p. 308
Symmetric and Alternating Groups over Hilbertian Fields	p. 315
GAR-Realizations	p. 321
Embedding Problems over Hilbertian Fields	p. 325
Finitely Generated Profinite Groups	p. 328
Abelian Extensions of Hilbertian Fields	p. 332
Regularity of Finite Groups over Complete Discrete Valued Fields	p. 334
Exercises	p. 335
Notes	p. 336
Free Profinite Groups	p. 338
The Rank of a Profinite Group	p. 338
Profinite Completions of Groups	p. 340
Formations of Finite Groups	p. 344
Free pro-C Groups	p. 346
Subgroups of Free Discrete Groups	p. 350
Open Subgroups of Free Profinite Groups	p. 358
An Embedding Property	p. 360
Exercises	p. 361
Notes	p. 362
The Haar Measure	p. 363
The Haar Measure of a Profinite Group	p. 363
Existence of the Haar Measure	p. 366
Independence	p. 370
Cartesian Product of Haar Measures	p. 376
The Haar Measure of the Absolute Galois Group	p. 378
The PAC Nullstellensatz	p. 380
The Bottom Theorem	p. 382
PAC Fields over Uncountable Hilbertian Fields	p. 386
On the Stability of Fields	p. 390
PAC Galois Extensions of Hilbertian Fields	p. 394
Algebraic Groups	p. 397
Exercises	p. 400
Notes	p. 401
Effective Field Theory and Algebraic Geometry	p. 403
Presented Rings and Fields	p. 403
Extensions of Presented Fields	p. 406
Galois Extensions of Presented Fields	p. 411
The Algebraic and Separable Closures of Presented Fields	p. 412
Constructive Algebraic Geometry	p. 413
Presented Rings and Constructible Sets	p. 422
Basic Normal Stratification	p. 425
Exercises	p. 427
Notes	p. 428
The Elementary Theory of e-Free PAC Fields	p. 429
N1-Saturated PAC Fields	p. 429
The Elementary Equivalence Theorem of N1-Saturated PAC Fields	p. 430
Elementary Equivalence of PAC Fields	p. 433
On e-Free PAC Fields	p. 436
The Elementary Theory of Perfect e-Free PAC Fields	p. 438
The Probable Truth of a Sentence	p. 440
Change of Base Field	p. 442
The Fields Ks(¿1,..., ¿e)	p. 444
The Transfer Theorem	p. 446
The Elementary Theory of Finite Fields	p. 448
Exercises	p. 451
Notes	p. 453
Problems of Arithmetical Geometry	p. 454
The Decomposition-Intersection Procedure	p. 454
Ci-Fields and Weakly Ci-Fields	p. 455
Perfect PAC Fields which are Ci	p. 460
The Existential Theory of PAC Fields	p. 462
Kronecker Classes of Number Fields	p. 463
Davenport's Problem	p. 467
On permutation Groups	p. 472
Schur's Conjecture	p. 479
Generalized Carlitz's Conjecture	p. 489
Exercises	p. 493
Notes	p. 495
Projective Groups and Frattini Covers	p. 497
The Frattini Groups of a Profinite Group	p. 497
Cartesian Squares	p. 499
On C Projective Groups	p. 502
Projective Groups	p. 506
Frattini Covers	p. 508
The Universal Frattini Cover	p. 513
Projective Pro-p-Groups	p. 515
Supernatural Numbers	p. 520
The Sylow Theorems	p. 522
On Complements of Normal Subgroups	p. 524
The Universal Frattini p-Cover	p. 528
Examples of Universal Frattini p-Covers	p. 532
The Special Linear Group SL(2, <$>{\cal Z}_p<$>)	p. 534
The General Linear Group GL(2, <$>{\cal Z}_p<$>)	p. 537
Exercises	p. 539
Notes	p. 542
PAC Fields and Projective Absolute Galois Groups	p. 544
Projective Groups as Absolute Galois Groups	p. 544
Countably Generated Projective Groups	p. 546
Perfect PAC Fields of Bounded Corank	p. 549
Basic Elementary Statements	p. 550
Reduction Steps	p. 554
Application of Ultraproducts	p. 558
Exercises	p. 561
Notes	p. 561
Frobenius Fields	p. 562
The Field Crossing Argument	p. 562
The Beckmann-Black Problem	p. 565
The Embedding Property and Maximal Frattini Covers	p. 567
The Smallest Embedding Cover of a Profinite Group	p. 569
A Decision Procedure	p. 574
Examples	p. 576
Non-projective Smallest Embedding Cover	p. 579
A Theorem of Iwasawa	p. 581
Free Profinite Groups of at most Countable Rank	p. 583
Application of the Nielsen-Schreier Formula	p. 586
Exercises	p. 591
Notes	p. 592
Free Profinite Groups of Infinite Rank	p. 594
Characterization of Free Profinite Groups by Embedding Problems	p. 595
Applications of Theorem 25.1.7	p. 601
The Pro-C Completion of a Free Discrete Group	p. 604
The Group Theoretic Diamond Theorem	p. 606
The Melnikov Group of a Profinite Group	p. 613
Homogeneous Pro-C Groups	p. 615
The S-rank of Closed Normal Subgroups	p. 620
Closed Normal Subgroups with a Basis Element	p. 623
Accessible Subgroups	p. 625
Notes	p. 633
Random Elements in Free Profinite Groups	p. 635
Random Elements in a Free Profinite Group	p. 635
Random Elements in Free pro-p Groups	p. 640
Random e-tuples in <$>\hat {\op Z}^n<$>	p. 642
On the Index of Normal Subgroups Generated by Random Elements	p. 646
Freeness of Normal Subgroups Generated by Random Elements	p. 651
Notes	p. 654
Omega-Free PAC Fields	p. 655
Model Companions	p. 655
The Model Companion in an Augmented Theory of Fields	p. 659
New Non-Classical Hilbertian Fields	p. 664
An abundance of ¿-Free PAC Fields	p. 667
Notes	p. 670
Undecidability	p. 671
Turing Machines	p. 671
Computation of Functions by Turing Machines	p. 672
Recursive Inseparability of Sets of Turing Machines	p. 676
The Predicate Calculus	p. 679
Undecidability in the Theory of Graphs	p. 682
Assigning Graphs to Profinite Groups	p. 687
The Graph Conditions	p. 688
Assigning Profinite Groups to Graphs	p. 690
Assigning Fields to Graphs	p. 694
Interpretation of the Theory of Graphs in the Theory of Fields	p. 694
Exercises	p. 697
Notes	p. 697
Algebraically Closed Fields with Distinguished Automorphisms	p. 698
The Base Field K	p. 698
Coding in PAC Fields with Monadic Quantifiers	p. 700
The Theory of Almost all ⟨<$>\tilde {K}<$>, ¿1, ..., ¿e⟩'s	p. 704
The Probability of Truth Sentences	p. 706
Galois Stratification	p. 708
The Artin Symbol	p. 708
Conjugacy Domains under Projection	p. 710
Normal Stratification	p. 715
Elimination of One Variable	p. 717
The Complete Elimination Procedure	p. 720
Model-Theoretic Applications	p. 722
A Limit of Theories	p. 725
Exercises	p. 726
Notes	p. 729
Galois Stratification over Finite Fields	p. 730
The Elementary Theory of Frobenius Fields	p. 730
The Elementary Theory of Finite Fields	p. 735
Near Rationality of the Zeta Function of a Galois Formula	p. 739
Exercises	p. 748
Notes	p. 750
Problems of Field Arithmetic	p. 751
Open Problems of the First Edition	p. 751
Open Problems of the Second Edition	p. 754
Open problems	p. 758
References	p. 761
Index	p. 780
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