Table of Contents - UCI Math

Table of Contents Chapter 1. Infinite Galois Theory and Profinite Groups 1.1 Inverse Limits . . . . . . . . . . . . . . . 1.2 Profinite Groups . . . . . . . . . . . . . . 1.3 Infinite Galois Theory . . . . . . . . . . . . 1.4 The p-adic Integers and the Pr¨ ufer Group . . . 1.5 The Absolute Galois Group of a Finite Field . . Exercises . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Valuations and Linear Disjointness . 2.1 Valuations, Places, and Valuation Rings . . 2.2 Discrete Valuations . . . . . . . . . . . 2.3 Extensions of Valuations and Places. . . . 2.4 Integral Extensions and Dedekind Domains 2.5 Linear Disjointness of Fields . . . . . . . 2.6 Separable, Regular, and Primary Extensions 2.7 The Imperfect Degree of a Field . . . . . 2.8 Derivatives . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . .

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19 19 21 24 30 34 38 44 48 50 51

Chapter 3. Algebraic Function Fields of One Variable . . 3.1 Function Fields of One Variable . . . . . . . . . 3.2 The Riemann-Roch Theorem . . . . . . . . . . 3.3 Holomorphy Rings . . . . . . . . . . . . . . . 3.4 Extensions of Function Fields . . . . . . . . . . 3.5 Completions . . . . . . . . . . . . . . . . . . 3.6 The Different . . . . . . . . . . . . . . . . . 3.7 Hyperelliptic Fields . . . . . . . . . . . . . . . 3.8 Hyperelliptic Fields with a Rational quadratic Subfield Exercises . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . .

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52 52 54 56 59 61 67 70 73 75 76

Chapter 4. The Riemann Hypothesis for Function Fields . 4.1 Class Numbers . . . . . . . . . . . . . . . . . 4.2 Zeta Functions . . . . . . . . . . . . . . . . . 4.3 Zeta Functions under Constant Field Extensions . . 4.4 The Functional Equation . . . . . . . . . . . . 4.5 The Riemann Hypothesis and Degree 1 Prime Divisors 4.6 Reduction Steps . . . . . . . . . . . . . . . . 4.7 An Upper Bound . . . . . . . . . . . . . . . . 4.8 A Lower Bound . . . . . . . . . . . . . . . .

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77 77 79 81 82 84 86 87 89

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1 1 4 9 12 15 16 18

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Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 5. Plane Curves . . . . . . . 5.1 Affine and Projective Plane Curves 5.2 Points and prime divisors . . . . 5.3 The Genus of a Plane Curve . . . 5.4 Points on a Curve over a Finite Field Exercises . . . . . . . . . . . . . Notes . . . . . . . . . . . . . .

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. 95 . 95 . 97 . 99 . 104 . 105 . 106

Chapter 6. The Chebotarev Density Theorem 6.1 Decomposition Groups . . . . . . . 6.2 The Artin Symbol over Global Fields . 6.3 Dirichlet Density . . . . . . . . . . 6.4 Function Fields . . . . . . . . . . 6.5 Number Fields . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . .

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107 107 111 113 114 121 129 130

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132 132 134 135 137 138 139 141 145 147 147 148

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149 149 152 154 156 157 159 160 161 162

Chapter 7. Ultraproducts . . . . . . . 7.1 First Order Predicate Calculus . . 7.2 Structures . . . . . . . . . . . 7.3 Models . . . . . . . . . . . . 7.4 Elementary Substructures . . . . 7.5 Ultrafilters . . . . . . . . . . 7.6 Regular Ultrafilters . . . . . . . 7.7 Ultraproducts . . . . . . . . . 7.8 Regular Ultraproducts . . . . . 7.9 Nonprincipal Ultraproducts of Finite Exercises . . . . . . . . . . . . . Notes . . . . . . . . . . . . . .

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91 93

Chapter 8. Decision Procedures . . . . . . . . 8.1 Deduction Theory . . . . . . . . . . . 8.2 G¨odel’s Completeness Theorem . . . . . 8.3 Primitive Recursive Functions . . . . . . 8.4 Primitive Recursive Relations . . . . . . 8.5 Recursive Functions . . . . . . . . . . 8.6 Recursive and Primitive Recursive Procedures 8.7 A Reduction Step in Decidability Procedures Exercises . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . .

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Chapter 9. Algebraically Closed Fields . 9.1 Elimination of quantifiers . . . . 9.2 A quantifiers elimination procedure 9.3 Effectiveness . . . . . . . . . . 9.4 Applications . . . . . . . . . . Exercises . . . . . . . . . . . . . Notes . . . . . . . . . . . . . .

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163 163 165 168 169 170 170

Chapter 10. Elements of Algebraic Geometry . . 10.1 Algebraic Sets . . . . . . . . . . . . 10.2 Varieties . . . . . . . . . . . . . . . 10.3 Substitutions in Irreducible Polynomials . 10.4 Rational Maps . . . . . . . . . . . . 10.5 Hyperplane Sections . . . . . . . . . . 10.6 Descent . . . . . . . . . . . . . . . 10.7 Projective Varieties . . . . . . . . . . 10.8 About the Language of Algebraic Geometry Exercises . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . .

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172 172 175 176 178 180 182 185 187 190 191

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192 192 193 199 201 203 206 211 216 217

Chapter 11. Pseudo Algebraically Closed Fields . . 11.1 PAC Fields . . . . . . . . . . . . . . . 11.2 Reduction to Plane Curves . . . . . . . . 11.3 The PAC Property is an Elementary Statement 11.4 PAC Fields of Positive Characteristic . . . 11.5 PAC Fields with Valuations . . . . . . . . 11.6 The Absolute Galois Group of a PAC Field . 11.7 A non-PAC Field K with Kins PAC . . . . Exercises . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . .

Chapter 12. Hilbertian Fields . . . . . . . . . . . . 12.1 Hilbert Sets and Reduction Lemmas . . . . . . 12.2 Hilbert Sets under Separable Algebraic Extensions 12.3 Purely Inseparable Extensions . . . . . . . . 12.4 Imperfect fields . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . .

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218 218 222 223 227 228 229

Chapter 13. The Classical Hilbertian Fields 13.1 Further Reduction . . . . . . . . 13.2 Function Fields over Infinite Fields 13.3 Global Fields . . . . . . . . . . 13.4 Hilbertian Rings . . . . . . . . 13.5 Hilbertianity via Coverings . . . .

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230 230 235 236 240 243

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13.6 Non-Hilbertian g-Hilbertian Fields 13.7 Twisted Wreath Products . . . 13.8 The Diamond Theorem . . . . 13.9 Weissauer’s Theorem . . . . . Exercises . . . . . . . . . . . . . Notes . . . . . . . . . . . . . .

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247 251 257 261 263 265

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266 266 267 269 271 273 274 275

Chapter 15. Nonstandard Approach to Hilbert’s Irreducibility Theorem 15.1 Criteria for Hilbertianity . . . . . . . . . . . . 15.2 Arithmetical Primes Versus Functional Primes . . 15.3 Fields with the Product Formula . . . . . . . . 15.4 Generalized Krull Domains . . . . . . . . . . . 15.5 Examples . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . .

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276 276 278 280 282 285 288 289

Chapter 16. Galois Groups over Hilbertian Fields . . . . . . 16.1 Galois Groups of Polynomials . . . . . . . . . . . . 16.2 Stable Polynomials . . . . . . . . . . . . . . . . 16.3 Regular Realization of Finite Abelian Groups . . . . . 16.4 Split Embedding Problems with Abelian Kernels . . . 16.5 Embedding Quadratic Extensions in Z/2n Z-extensions . 16.6 Zp -Extensions of Hilbertian Fields . . . . . . . . . . 16.7 Symmetric and Alternating Groups over Hilbertian Fields 16.8 GAR-Realizations . . . . . . . . . . . . . . . . . 16.9 Embedding Problems over Hilbertian Fields . . . . . 16.10 Finitely Generated Profinite Groups . . . . . . . . 16.11 Abelian Extensions of Hilbertian Fields . . . . . . . 16.12 Regularity of Finite Groups over Complete Discrete Valued Fields . . Exercises . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . .

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290 290 293 297 301 305 307 314 320 324 327 331

Chapter 14. Nonstandard Structures . 14.1 Higher Order Predicate Calculus 14.2 Enlargements . . . . . . . . 14.3 Concurrent Relations . . . . 14.4 The Existence of Enlargements 14.5 Examples . . . . . . . . . . Exercises . . . . . . . . . . . . Notes . . . . . . . . . . . . .

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. . 333 . . 334 . . 335

Chapter 17. Free Profinite Groups . . . . . . . . . . . . . . . 337 17.1 The Rank of a Profinite Group . . . . . . . . . . . . . 337

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17.2 Profinite Completions of Groups . . . 17.3 Formations of Finite Groups . . . . . 17.4 Free pro-C Groups . . . . . . . . . . 17.5 Subgroups of Free Discrete Groups . . 17.6 Open Subgroups of Free Profinite Groups 17.7 An Embedding Property . . . . . . . Exercises . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . .

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339 342 345 349 357 359 360 361

Chapter 18. The Haar Measure . . . . . . . . . . 18.1 The Haar Measure of a Profinite Group . . . 18.2 Existence of the Haar Measure . . . . . . . 18.3 Independence . . . . . . . . . . . . . . . 18.4 Cartesian Product of Haar Measures . . . . . 18.5 The Haar Measure of the Absolute Galois Group 18.6 The PAC Nullstellensatz . . . . . . . . . . 18.7 The Bottom Theorem . . . . . . . . . . . 18.8 PAC Fields over Uncountable Hilbertian Fields 18.9 On the Stability of Fields . . . . . . . . . . 18.10 PAC Galois Extensions of Hilbertian Fields . 18.11 Algebraic Groups . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . .

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362 362 365 369 375 377 379 381 385 389 393 396 398 399

Chapter 19. Effective Field Theory and Algebraic Geometry . 19.1 Presented Rings and Fields . . . . . . . . . . . . . 19.2 Extensions of Presented Fields . . . . . . . . . . . 19.3 Galois Extensions of Presented Fields . . . . . . . . 19.4 The Algebraic and Separable Closures of Presented Fields 19.5 Constructive Algebraic Geometry . . . . . . . . . . 19.6 Presented Rings and Constructible Sets . . . . . . . 19.7 Basic Normal Stratification . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . .

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401 401 404 409 410 411 420 423 425 426

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Chapter 20. The Elementary Theory of e-Free PAC Fields 20.1 ℵ1 -Saturated PAC Fields . . . . . . . . . . . . 20.2 The Elementary Equivalence Theorem of ℵ1 -Saturated PAC Fields . . . 20.3 Elementary Equivalence of PAC Fields . . . . . . 20.4 On e-Free PAC Fields . . . . . . . . . . . . . 20.5 The Elementary Theory of Perfect e-Free PAC Fields 20.6 The Probable Truth of a Sentence . . . . . . . . 20.7 Change of Base Field . . . . . . . . . . . . . 20.8 The Fields Ks (σ1 , . . . , σe ) . . . . . . . . . . .

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428 431 434 436 438 440 442

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20.9 The Transfer Theorem . . . . . . . . 20.10 The Elementary Theory of Finite Fields Exercises . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . .

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444 446 449 451

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452 452 453 458 460 461 465 470 477 487 490 493

Chapter 22. Projective Groups and Frattini Covers 22.1 The Frattini Groups of a Profinite Group . 22.2 Cartesian Squares . . . . . . . . . . . 22.3 On C-Projective Groups . . . . . . . . 22.4 Projective Groups . . . . . . . . . . . 22.5 Frattini Covers . . . . . . . . . . . . 22.6 The Universal Frattini Cover . . . . . . 22.7 Projective Pro-p-Groups . . . . . . . . 22.8 Supernatural Numbers . . . . . . . . . 22.9 The Sylow Theorems . . . . . . . . . 22.10 On Complements of Normal Subgroups . 22.11 The Universal Frattini p-Cover . . . . . 22.12 Examples of Universal Frattini p-Covers . 22.13 The Special Linear Group SL(2, Zp ) . . 22.14 The General Linear Group GL(2, Zp ) . . Exercises . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . .

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494 494 496 499 503 505 510 512 517 519 521 525 529 531 534 536 539

Chapter 23. PAC Fields and Projective Absolute Galois Groups 23.1 Projective Groups as Absolute Galois Groups . . . . . 23.2 Countably Generated Projective Groups . . . . . . . 23.3 Perfect PAC Fields of Bounded Corank . . . . . . . 23.4 Basic Elementary Statements . . . . . . . . . . . . 23.5 Reduction Steps . . . . . . . . . . . . . . . . . . 23.6 Application of Ultraproducts . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . .

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541 541 543 546 547 551 555 558 558

Chapter 21. Problems of Arithmetical Geometry 21.1 The Decomposition-Intersection Procedure 21.2 Ci -Fields and Weakly Ci -Fields . . . . 21.3 Perfect PAC Fields which are Ci . . . 21.4 The Existential Theory of PAC Fields . 21.5 Kronecker Classes of Number Fields . . 21.6 Davenport’s Problem . . . . . . . . 21.7 On permutation Groups . . . . . . . 21.8 Schur’s Conjecture . . . . . . . . . . 21.9 Carlitz-Wan Conjecture . . . . . . . Exercises . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . .

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Chapter 24. Frobenius Fields . . . . . . . . . . . . . . 24.1 The Field Crossing Argument . . . . . . . . . . . 24.2 The Beckmann-Black Problem . . . . . . . . . . 24.3 The Embedding Property and Maximal Frattini Covers 24.4 The Smallest Embedding Cover of a Profinite Group . 24.5 A Decision Procedure . . . . . . . . . . . . . . 24.6 Examples . . . . . . . . . . . . . . . . . . . . 24.7 Non-projective Smallest Embedding Cover . . . . . 24.8 A Theorem of Iwasawa . . . . . . . . . . . . . . 24.9 Free Profinite Groups of at most Countable Rank . . 24.10 Application of the Nielsen-Schreier Formula . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . .

. . . 559 . . . 559 . . . 562 . . 564 . . . 566 . . . 571 . . . 573 . . . 576 . . . 578 . . . 580 . . . 583 . . . 588 . . . 589

Chapter 25. Free Profinite Groups of Infinite Rank . . 25.1 Characterization of Free Profinite Groups by Embedding Problems 25.2 Applications of Theorem 25.1.7 . . . . . . . . 25.3 The Pro-C Completion of a Free Discrete Group . 25.4 The Group Theoretic Diamond Theorem . . . . 25.5 The Melnikov Group of a Profinite Group . . . 25.6 Homogeneous Pro-C Groups . . . . . . . . . 25.7 The S-rank of Closed Normal Subgroups . . . . 25.8 Closed Normal Subgroups with a Basis Element . 25.9 Accessible Subgroups . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . .

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592 598 601 603 610 612 617 620 622 630

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632 632 637 639

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Chapter 26. Random Elements in Free Profinite Groups . 26.1 Random Elements in a Free Profinite Group . . . 26.2 Random Elements in Free pro-p Groups . . . . . ˆn . . . . . . . . . . . . . 26.3 Random e-tuples in Z 26.4 On the Index of Normal Subgroups Generated by Random Elements . 26.5 Freeness of Normal Subgroups Generated by Random Elements Notes . . . . . . . . . . . . . . . . . . . . . . Chapter 27. Omega-Free PAC Fields . . . . . . . . . 27.1 Model Companions . . . . . . . . . . . . . 27.2 The Model Companion in an Augmented Theory of 27.3 New Non-Classical Hilbertian Fields . . . . . . 27.4 An abundance of ω-Free PAC Fields . . . . . . Notes . . . . . . . . . . . . . . . . . . . . .

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Chapter 28. Undecidability . . . . . . . . . . . . . . 28.1 Turing Machines . . . . . . . . . . . . . . . 28.2 Computation of Functions by Turing Machines . . 28.3 Recursive Inseparability of Sets of Turing Machines 28.4 The Predicate Calculus . . . . . . . . . . . . 28.5 Undecidability in the Theory of Graphs . . . . . 28.6 Assigning Graphs to Profinite Groups . . . . . . 28.7 The Graph Conditions . . . . . . . . . . . . . 28.8 Assigning Profinite Groups to Graphs . . . . . . 28.9 Assigning Fields to Graphs . . . . . . . . . . . 28.10 Interpretation of the Theory of Graphs in the Theory of Fields Exercises . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . .

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668 668 669 673 676 679 684 685 687 691

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Chapter 29. Algebraically Closed Fields with Distinguished Automorphisms 29.1 The Base Field K . . . . . . . . . . . . . . . . . 29.2 Coding in PAC Fields with Monadic Quantifiers . . . . ˜ σ1 , . . . , σe i’s 29.3 The Theory of Almost all hK, . . . . . . 29.4 The Probability of Truth Sentences . . . . . . . . .

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. 695 695 697 701 703

Chapter 30. Galois Stratification . . . . 30.1 The Artin Symbol . . . . . . . . 30.2 Conjugacy Domains under Projection 30.3 Normal Stratification . . . . . . 30.4 Elimination of One Variable . . . 30.5 The Complete Elimination Procedure 30.6 Model-Theoretic Applications . . . 30.7 A Limit of Theories . . . . . . . Exercises . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . .

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705 705 707 712 714 717 719 722 723 726

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Chapter 31. Galois Stratification over Finite Fields . 31.1 The Elementary Theory of Frobenius Fields . 31.2 The Elementary Theory of Finite Fields . . . 31.3 Near Rationality of the Zeta Function of a Galois Exercises . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . .

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. . 727 . . 727 . . 732 . 736 . . 745 . . 747

Chapter 32. Problems of Field Arithmetic . . . . . . . . . . . 748 32.1 Open Problems of the First Edition . . . . . . . . . . . 748 32.2 Open Problems . . . . . . . . . . . . . . . . . . . . 751 References Index

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