Field Arithmetic (eBook)

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 6
Introduction to the Third Edition 15
Introduction to the First Edition 19
Notation and Convention 23
Chapter 1. Infinite Galois Theory and Profinite Groups 24
1.1 Inverse Limits 24
1.2 Profinite Groups 27
1.3 Infinite Galois Theory 32
1.4 The p-adic Integers and the Pr¨ufer Group 35
1.5 The Absolute Galois Group of a Finite Field 38
Chapter 2. Valuations and Linear Disjointness 42
2.1 Valuations, Places, and Valuation Rings 42
2.2 Discrete Valuations 44
2.3 Extensions of Valuations and Places 47
2.4 Integral Extensions and Dedekind Domains 53
2.5 Linear Disjointness of Fields 57
2.6 Separable, Regular, and Primary Extensions 61
2.7 The Imperfect Degree of a Field 67
2.8 Derivatives 71
Chapter 3. Algebraic Function Fields of One Variable 75
3.1 Function Fields of One Variable 75
3.2 The Riemann-Roch Theorem 77
3.3 Holomorphy Rings 79
3.4 Extensions of Function Fields 82
3.5 Completions 84
3.6 The Different 90
3.7 Hyperelliptic Fields 93
3.8 Hyperelliptic Fields with a Rational Quadratic Subfield 96
Exercises 98
Notes 99
Chapter 4. The Riemann Hypothesis for Function Fields 100
4.1 Class Numbers 100
4.2 Zeta Functions 102
4.3 Zeta Functions under Constant Field Extensions 104
4.4 The Functional Equation 105
4.5 The Riemann Hypothesis and Degree 1 Prime Divisors 107
4.6 Reduction Steps 109
4.7 An Upper Bound 110
4.8 A Lower Bound 112
Exercises 114
Notes 116
Chapter 5. Plane Curves 118
5.1 A ne and Projective Plane Curves 118
5.2 Points and Prime Divisors 120
5.3 The Genus of a Plane Curve 122
5.4 Points on a Curve over a Finite Field 127
Exercises 128
Notes 129
Chapter 6. The Chebotarev Density Theorem 130
6.1 Decomposition Groups 130
6.2 The Artin Symbol over Global Fields 134
6.3 Dirichlet Density 136
6.4 Function Fields 138
Exercises 152
Notes 153
Chapter 7. Ultraproducts 155
7.1 First Order Predicate Calculus 155
7.2 Structures 157
7.3 Models 158
7.4 Elementary Substructures 160
7.5 Ultrafilters 161
7.6 Regular Ultrafilters 162
7.7 Ultraproducts 164
7.8 Regular Ultraproducts 168
7.9 Nonprincipal Ultraproducts of Finite Fields 170
Exercises 170
Notes 171
Chapter 8. Decision Procedures 172
8.1 Deduction Theory 172
8.2 Gödel’s Completeness Theorem 175
8.3 Primitive Recursive Functions 177
8.4 Primitive Recursive Relations 179
8.5 Recursive Functions 180
8.6 Recursive and Primitive Recursive Procedures 182
8.7 A Reduction Step in Decidability Procedures 183
Exercises 184
Notes 185
Chapter 9. Algebraically Closed Fields 186
9.1 Elimination of Quantifiers 186
9.2 A Quantifiers Elimination Procedure 188
9.3 E ectiveness 191
9.4 Applications 192
Exercises 193
Notes 193
Chapter 10. Elements of Algebraic Geometry 195
10.1 Algebraic Sets 195
10.2 Varieties 198
10.3 Substitutions in Irreducible Polynomials 199
10.4 Rational Maps 201
10.5 Hyperplane Sections 203
10.6 Descent 205
10.7 Projective Varieties 208
10.8 About the Language of Algebraic Geometry 210
Notes 214
Chapter 11. Pseudo Algebraically Closed Fields 215
11.1 PAC Fields 215
11.2 Reduction to Plane Curves 216
11.3 The PAC Property is an Elementary Statement 222
11.4 PAC Fields of Positive Characteristic 224
11.5 PAC Fields with Valuations 226
11.6 The Absolute Galois Group of a PAC Field 230
11.7 A non-PAC Field K with Kins PAC 234
Exercises 240
Notes 241
Chapter 12. Hilbertian Fields 242
12.1 Hilbert Sets and Reduction Lemmas 242
12.2 Hilbert Sets under Separable Algebraic Extensions 246
12.3 Purely Inseparable Extensions 247
12.4 Imperfect fields 251
Exercises 252
Notes 253
Chapter 13. The Classical Hilbertian Fields 254
13.1 Further Reduction 254
13.2 Function Fields over Infinite Fields 259
13.3 Global Fields 260
13.4 Hilbertian Rings 264
13.5 Hilbertianity via Coverings 267
13.6 Non-Hilbertian g-Hilbertian Fields 271
13.7 Twisted Wreath Products 275
13.8 The Diamond Theorem 281
13.9 Weissauer’s Theorem 285
Exercises 287
Notes 289
Chapter 14. Nonstandard Structures 290
14.1 Higher Order Predicate Calculus 290
14.2 Enlargements 291
14.3 Concurrent Relations 293
14.4 The Existence of Enlargements 295
14.5 Examples 297
Exercises 298
Notes 299
Chapter 15. Nonstandard Approach to Hilbert’s Irreducibility Theorem 300
15.1 Criteria for Hilbertianity 300
15.2 Arithmetical Primes Versus Functional Primes 302
15.3 Fields with the Product Formula 304
15.4 Generalized Krull Domains 306
15.5 Examples 309
Exercises 312
Notes 313
Chapter 16. Galois Groups over Hilbertian Fields 314
16.1 Galois Groups of Polynomials 314
16.2 Stable Polynomials 317
16.3 Regular Realization of Finite Abelian Groups 321
16.4 Split Embedding Problems with Abelian Kernels 325
16.5 Embedding Quadratic Extensions in Z/2nZ-Extensions 329
16.6 Zp-Extensions of Hilbertian Fields 331
16.7 Symmetric and Alternating Groups over Hilbertian Fields 338
16.8 GAR-Realizations 344
16.9 Embedding Problems over Hilbertian Fields 348
16.10 Finitely Generated Profinite Groups 351
16.11 Abelian Extensions of Hilbertian Fields 355
16.12 Regularity of Finite Groups over Complete Discrete Valued Fields 357
Exercises 358
Notes 359
Chapter 17. Free Profinite Groups 361
17.1 The Rank of a Profinite Group 361
17.2 Profinite Completions of Groups 363
17.3 Formations of Finite Groups 367
17.4 Free pro-C Groups 369
17.5 Subgroups of Free Discrete Groups 373
17.6 Open Subgroups of Free Profinite Groups 381
17.7 An Embedding Property 383
Exercises 384
Notes 385
Chapter 18. The Haar Measure 386
18.1 The Haar Measure of a Profinite Group 386
18.2 Existence of the Haar Measure 389
18.3 Independence 393
18.4 Cartesian Product of Haar Measures 399
18.5 The Haar Measure of the Absolute Galois Group 401
18.6 The PAC Nullstellensatz 403
18.7 The Bottom Theorem 405
18.8 PAC Fields over Uncountable Hilbertian Fields 409
18.9 On the Stability of Fields 413
18.10 PAC Galois Extensions of Hilbertian Fields 417
18.11 Algebraic Groups 420
Exercises 423
Notes 424
Chapter 19. E ective Field Theory and Algebraic Geometry 426
19.1 Presented Rings and Fields 426
19.2 Extensions of Presented Fields 429
19.3 Galois Extensions of Presented Fields 434
19.4 The Algebraic and Separable Closures of Presented Fields 435
19.5 Constructive Algebraic Geometry 436
19.6 Presented Rings and Constructible Sets 445
19.7 Basic Normal Stratification 448
Exercises 450
Notes 451
Chapter 20. The Elementary Theory of e-Free PAC Fields 452
20.1 @1-Saturated PAC Fields 452
20.2 The Elementary Equivalence Theorem of @1-Saturated PAC Fields 453
20.3 Elementary Equivalence of PAC Fields 456
20.4 On e-Free PAC Fields 459
20.5 The Elementary Theory of Perfect e-Free PAC Fields 461
20.6 The Probable Truth of a Sentence 463
20.7 Change of Base Field 465
20.8 The Fields Ks( 1, . . . , e) 467
20.9 The Transfer Theorem 469
20.10 The Elementary Theory of Finite Fields 471
Exercises 474
Notes 476
Chapter 21. Problems of Arithmetical Geometry 477
21.1 The Decomposition-Intersection Procedure 477
21.2 Ci-Fields and Weakly Ci-Fields 478
21.3 Perfect PAC Fields which are Ci 483
21.4 The Existential Theory of PAC Fields 485
21.5 Kronecker Classes of Number Fields 486
21.6 Davenport’s Problem 490
21.7 On Permutation Groups 495
21.8 Schur’s Conjecture 502
21.9 Generalized Carlitz’s Conjecture 512
Exercises 516
Notes 518
Chapter 22. Projective Groups and Frattini Covers 520
22.1 The Frattini Group of a Profinite Group 520
22.2 Cartesian Squares 522
22.3 On C-Projective Groups 525
22.4 Projective Groups 529
22.5 Frattini Covers 531
22.6 The Universal Frattini Cover 536
22.7 Projective Pro-p-Groups 538
22.8 Supernatural Numbers 543
22.9 The Sylow Theorems 545
22.10 On Complements of Normal Subgroups 547
22.11 The Universal Frattini p-Cover 551
22.12 Examples of Universal Frattini p-covers 555
22.13 The Special Linear Group SL(2, Zp) 557
22.14 The General Linear Group GL(2,Zp) 560
Exercises 562
Notes 565
Chapter 23. PAC Fields and Projective Absolute Galois Groups 567
23.1 Projective Groups as Absolute Galois Groups 567
23.2 Countably Generated Projective Groups 569
23.3 Perfect PAC Fields of Bounded Corank 572
23.4 Basic Elementary Statements 573
23.5 Reduction Steps 577
23.6 Application of Ultraproducts 581
Exercises 584
Notes 584
Chapter 24. Frobenius Fields 585
24.1 The Field Crossing Argument 585
24.2 The Beckmann-Black Problem 588
24.3 The Embedding Property and Maximal Frattini Covers 590
24.4 The Smallest Embedding Cover of a Profinite Group 592
24.5 A Decision Procedure 597
24.6 Examples 599
24.7 Non-projective Smallest Embedding Cover 602
24.8 A Theorem of Iwasawa 604
24.9 Free Profinite Groups of at most Countable Rank 606
24.10 Application of the Nielsen-Schreier Formula 609
Exercises 614
Notes 615
Chapter 25. Free Profinite Groups of Infinite Rank 617
25.1 Characterization of Free Profinite Groups by Embedding Problems 618
25.2 Applications of Theorem 25.1.7 624
25.3 The Pro-C Completion of a Free Discrete Group 627
25.4 The Group Theoretic Diamond Theorem 629
25.5 The Melnikov Group of a Profinite Group 636
25.6 Homogeneous Pro-C Groups 638
25.7 The S-rank of Closed Normal Subgroups 643
25.8 Closed Normal Subgroups with a Basis Element 646
25.9 Accessible Subgroups 648
Notes 656
Chapter 26. Random Elements in Profinite Groups 658
26.1 Random Elements in a Free Profinite Group 658
26.2 Random Elements in Free pro-p Groups 663
26.3 Random e-tuples in Z 665
26.4 On the Index of Normal Subgroups Generated by Random Elements 669
26.5 Freeness of Normal Subgroups Generated by Random Elements 674
Notes 677
Chapter 27. Omega-free PAC Fields 678
27.1 Model Companions 678
27.2 The Model Companion in an Augmented Theory of Fields 682
27.3 New Non-Classical Hilbertian Fields 687
27.4 An Abundance of !-Free PAC Fields 690
Notes 693
Chapter 28. Undecidability 694
28.1 Turing Machines 694
28.2 Computation of Functions by Turing Machines 695
28.3 Recursive Inseparability of Sets of Turing Machines 699
28.4 The Predicate Calculus 702
28.5 Undecidability in the Theory of Graphs 705
28.6 Assigning Graphs to Profinite Groups 710
28.7 The Graph Conditions 711
28.8 Assigning Profinite Groups to Graphs 713
28.9 Assigning Fields to Graphs 717
28.10 Interpretation of the Theory of Graphs in the Theory of Fields 717
Exercises 720
Notes 720
Chapter 29. Algebraically Closed Fields with Distinguished Automorphisms 721
29.1 The Base Field K 721
29.2 Coding in PAC Fields with Monadic Quantifiers 723
29.3 The Theory of Almost all 727
29.4 The Probability of Truth Sentences 729
Chapter 30. Galois Stratification 731
30.1 The Artin Symbol 731
30.2 Conjugacy Domains under Projections 733
30.3 Normal Stratification 738
30.4 Elimination of One Variable 740
30.5 The Complete Elimination Procedure 743
30.6 Model-Theoretic Applications 745
30.7 A Limit of Theories 748
Exercises 749
Notes 752
Chapter 31. Galois Stratification over Finite Fields 753
31.1 The Elementary Theory of Frobenius Fields 753
31.2 The Elementary Theory of Finite Fields 758
31.3 Near Rationality of the Zeta Function of a Galois Formula 762
Exercises 771
Notes 773
Chapter 32. Problems of Field Arithmetic 774
32.1 Open Problems of the First Edition 774
32.2 Open Problems of the Second Edition 777
32.3 Open Problems 781
References 784
Index 803