Field Arithmetic

Main subject categories: • Field arithmetic • Algebra • Number theory • Algebraic geometry • AnalysisThis book uses algebraic tools to study the elementary properties of classes of fields and related algorithmic problems. The first part covers foundational material on infinite Galois theory, profinite groups, algebraic function fields in one variable and plane curves. It provides complete and elementary proofs of the Chebotarev density theorem and the Riemann hypothesis for function fields, together with material on ultraproducts, decision procedures, the elementary theory of algebraically closed fields, undecidability and nonstandard model theory, including a nonstandard proof of Hilbert's irreducibility theorem. The focus then turns to the study of pseudo algebraically closed (PAC) fields, related structures and associated decidability and undecidability results. PAC fields (fields K with the property that every absolutely irreducible variety over K has a rational point) first arose in the elementary theory of finite fields and have deep connections with number theory.This fourth edition substantially extends, updates and clarifies the previous editions of this celebrated book, and includes a new chapter on Hilbertian subfields of Galois extensions. Almost every chapter concludes with a set of exercises and bibliographical notes. An appendix presents a selection of open research problems.Drawing from a wide literature at the interface of logic and arithmetic, this detailed and self-contained text can serve both as a textbook for graduate courses and as an invaluable reference for seasoned researchers. Preface to the Fourth Edition Main New Results Detailed List of New Results Problems of Field Arithmetic Structural Changes Typing Programs Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Notation and Conventions 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 Prüfer Group 1.5 The Absolute Galois Group of a Finite Field Exercises Notes Chapter 2 Valuations 2.1 Valuations, Places, and Valuation Rings 2.2 Discrete Valuations 2.3 Extensions of Valuations and Places 2.4 Galois Extensions 2.5 Integral Extensions and Dedekind Domains Exercises Chapter 3 Linear Disjointness 3.1 Linear Disjointness of Fields 3.2 Purely Transcendental Extensions 3.3 Separable Extensions 3.4 Regular Extensions 3.5 Primary Extensions 3.6 The Imperfect Degree of a Field 3.7 Derivatives Exercises Chapter 4 Algebraic Function Fields of One Variable 4.1 Function Fields of One Variable 4.2 The Riemann–Roch Theorem 4.3 Holomorphy Rings 4.4 Extensions of Function Fields 4.5 Completions 4.6 The Different 4.7 Hyperelliptic Fields 4.8 Hyperelliptic Fields with a Rational Quadratic Subfield Exercises Notes Chapter 5 The Riemann Hypothesis for Function Fields 5.1 Class Numbers 5.2 Zeta Functions 5.3 Zeta Functions under Constant Field Extensions 5.4 The Functional Equation 5.5 The Riemann Hypothesis and Degree 1 Prime Divisors 5.6 Reduction Steps 5.7 An Upper Bound 5.8 A Lower Bound Exercises Notes Chapter 6 Plane Curves 6.1 Affine and Projective Plane Curves 6.2 Points and Prime Divisors 6.3 The Genus of a Plane Curve 6.4 Points on a Curve over a Finite Field Exercises Notes Chapter 7 The Chebotarev Density Theorem 7.1 Decomposition Groups 7.2 The Artin Symbol over Global Fields 7.3 Dirichlet Density 7.4 Function Fields 7.5 Number Fields Exercises Notes Chapter 8 Ultraproducts 8.1 First Order Predicate Calculus 8.2 Structures 8.3 Models 8.4 Elementary Substructures 8.5 Ultrafilters 8.6 Ultraproducts 8.7 Regular Ultrafilters 8.8 Regular Ultraproducts 8.9 Nonprincipal Ultraproducts of Finite Fields Exercises Notes Chapter 9 Decision Procedures 9.1 Deduction Theory 9.2 Gödel’s Completeness Theorem 9.3 Primitive Recursive Functions 9.4 Primitive Recursive Relations 9.5 Recursive Functions 9.6 Recursive and Primitive Recursive Procedures 9.7 A Reduction Step in Decidability Procedures Exercises Notes Chapter 10 Algebraically Closed Fields 10.1 Elimination of Quantifiers 10.2 A Quantifier Elimination Procedure 10.3 Effectiveness 10.4 Applications Exercises Notes Chapter 11 Elements of Algebraic Geometry 11.1 Algebraic Sets 11.2 Varieties 11.3 Substitutions in Irreducible Polynomials 11.4 Rational Maps 11.5 Hyperplane Sections 11.6 Descent 11.7 Projective Varieties 11.8 About the Language of Algebraic Geometry Exercises Notes Chapter 12 Pseudo Algebraically Closed Fields 12.1 PAC Fields 12.2 Reduction to Plane Curves 12.3 The PAC Property is an Elementary Statement 12.4 PAC Fields of Positive Characteristic 12.5 PAC Fields with Valuations 12.6 The Absolute Galois Group of a PAC Field 12.7 A non-PAC Field K with Kins PAC Exercises Notes Chapter 13 Hilbertian Fields 13.1 Hilbert Sets and Reduction Lemmas 13.2 Hilbert Sets under Separable Algebraic Extensions 13.3 Purely Inseparable Extensions 13.4 Imperfect Fields Exercises Notes Chapter 14 The Classical Hilbertian Fields 14.1 Further Reduction 14.2 Function Fields Over Infinite Fields 14.3 Global Fields 14.4 Hilbertian Rings 14.5 Hilbertianity via Coverings 14.6 Non-Hilbertian g-Hilbertian Fields Exercises Notes Chapter 15 The Diamond Theorem 15.1 Twisted Wreath Products 15.2 The Diamond Theorem 15.3 Weissauer’s Theorem Exercises Notes Chapter 16 Nonstandard Structures 16.1 Higher Order Predicate Calculus 16.2 Enlargements 16.3 Concurrent Relations 16.4 The Existence of Enlargements 16.5 Examples Exercises Notes Chapter 17 The Nonstandard Approach to Hilbert’s Irreducibility Theorem 17.1 Criteria for Hilbertianity 17.2 Arithmetical Primes Versus Functional Primes 17.3 Fields with the Product Formula 17.4 Generalized Krull Domains 17.5 Examples Exercises Notes Chapter 18 Galois Groups over Hilbertian Fields 18.1 Galois Groups of Polynomials 18.2 Stable Polynomials 18.3 Regular Realization of Finite Abelian Groups 18.4 Split Embedding Problems with Abelian Kernels 18.5 Embedding Quadratic Extensions in Z/2nZ-Extensions 18.6 Zp-Extensions of Hilbertian Fields 18.7 Symmetric and Alternating Groups over Hilbertian Fields 18.8 GAR-Realizations 18.9 Embedding Problems over Hilbertian Fields 18.10 Regularity of Finite Groups over Complete Discrete-Valued Fields Exercises Notes Chapter 19 Small Profinite Groups 19.1 Finitely Generated Profinite Groups 19.2 Abelian Extensions of Hilbertian Fields Exercises Notes Chapter 20 Free Profinite Groups 20.1 The Rank of a Profinite Group 20.2 Profinite Completions of Groups 20.3 Formations of Finite Groups 20.4 Free pro-C Groups 20.5 Subgroups of Free Discrete Groups 20.6 Open Subgroups of Free Profinite Groups 20.7 An Embedding Property Exercises Notes Chapter 21 The Haar Measure 21.1 The Haar Measure of a Profinite Group 21.2 Existence of the Haar Measure 21.3 Independence 21.4 Cartesian Product of Haar Measures 21.5 The Haar Measure of the Absolute Galois Group 21.6 The PAC Nullstellensatz 21.7 Baire’s Theorem 21.8 The Bottom Theorem 21.9 Triviality of a Group of Automorphisms 21.10 PAC Fields over Uncountable Hilbertian Fields 21.11 On the Stability of Fields 21.12 PAC Galois Extensions of Hilbertian Fields 21.13 Algebraic Groups Exercises Notes Chapter 22 Effective Field Theory and Algebraic Geometry 22.1 Presented Rings and Fields 22.2 Extensions of Presented Fields 22.3 Galois Extensions of Presented Fields 22.4 The Algebraic and Separable Closures of Presented Fields 22.5 Constructive Algebraic Geometry 22.6 Presented Rings and Constructible Sets 22.7 Basic Normal Stratification Exercises Notes Chapter 23 The Elementary Theory of e-Free PAC Fields 23.1 א1-Saturated PAC Fields 23.2 The Elementary Equivalence Theorem of א1-Saturated PAC Fields 23.3 Elementary Equivalence of PAC Fields 23.4 On On e-Free PAC Fields 23.5 The Elementary Theory of Perfect e-Free PAC Fields 23.6 The Probable Truth of a Sentence 23.7 Change of Base Field 23.8 The Fields Ksep(σ1, . . . , σe) 23.9 The Transfer Theorem 23.10 The Elementary Theory of Finite Fields Exercises Notes Chapter 24 Problems of Arithmetical Geometry 24.1 The Decomposition-Intersection Procedure 24.2 Ci-Fields and Weakly Ci-Fields 24.3 Perfect PAC Fields which are Ci 24.4 The Existential Theory of PAC Fields 24.5 Kronecker Classes of Number Fields 24.6 Davenport’s Problem 24.7 On Permutation Groups 24.8 Schur’s Conjecture 24.9 The Generalized Carlitz Conjecture Exercises Notes Chapter 25 Projective Groups and Frattini Covers 25.1 The Frattini Group of a Profinite Group 25.2 Cartesian Squares 25.3 On On C-Projective Groups 25.4 Projective Groups 25.5 Free Products of Finitely many Profinite Groups 25.6 Frattini Covers 25.7 The Universal Frattini Cover 25.8 Projective Pro- p-Groups 25.9 Supernatural Numbers 25.10 The Sylow Theorems 25.11 On Complements of Normal Subgroups 25.12 The Universal Frattini p-Cover 25.13 Examples of Universal Frattini p-covers 25.14 The Special Linear Group SL(2, Zp) 25.15 The General Linear Group GL(2, Zp) 25.16 Absolute Galois Groups Exercises Notes Chapter 26 PAC Fields and Projective Absolute Galois Groups 26.1 Projective Groups as Absolute Galois Groups 26.2 Countably Generated Projective Groups 26.3 Perfect PAC Fields of Bounded Corank 26.4 Basic Elementary Statements 26.5 Reduction Steps 26.6 Application of Ultraproducts Exercises Notes Chapter 27 Frobenius Fields 27.1 The Field Crossing Argument 27.2 The Beckmann–Black Problem 27.3 The Embedding Property and Maximal Frattini Covers 27.4 The Smallest Embedding Cover of a Profinite Group 27.5 A Decision Procedure 27.6 Examples 27.7 Non-projective Smallest Embedding Cover 27.8 A Theorem of Iwasawa 27.9 Free Profinite Groups of Countable Rank 27.10 Application of the Nielsen–Schreier Formula Exercises Notes Chapter 28 Free Profinite Groups of Infinite Rank 28.1 Characterization of Free Profinite Groups by Embedding Problems 28.2 Applications of Theorem 28.1.7 28.3 The Pro-Completion of a Free Discrete Group 28.4 The Group Theoretic Diamond Theorem 28.5 The Melnikov Group of a Profinite Group 28.6 Homogeneous Pro-C Groups 28.7 The The S-rank of Closed Normal Subgroups 28.8 Closed Normal Subgroups with a Basis Element 28.9 Accessible Subgroups Exercises Notes Chapter 29 Random Elements in Profinite Groups 29.1 Random Elements in a Free Profinite Group 29.2 Random Elements in Free pro-p Groups 29.3 Random Ẑn 29.4 The Golod–Shafarevich Inequality 29.5 On the Index of Normal Subgroups Generated by Random Elements 29.6 Freeness of Normal Subgroups Generated by Random Elements Notes Chapter 30 Omega-free PAC Fields 30.1 Model Companions 30.2 The Model Companion in an Augmented Theory of Fields 30.3 New Non-Classical Hilbertian Fields 30.4 An Abundance of ω-Free PAC Fields Notes Chapter 31 Hilbertian Subfields of Galois Extensions 31.1 Small Extensions 31.2 Auxiliary Results 31.3 The Main Result Notes Chapter 32 Undecidability 32.1 Turing Machines 32.2 Computation of Functions by Turing Machines 32.3 Recursive Inseparability of Sets of Turing Machines 32.4 The Predicate Calculus 32.5 Undecidability in the Theory of Graphs 32.6 Assigning Graphs to Profinite Groups 32.7 The Graph Conditions 32.8 Assigning Profinite Groups to Graphs 32.9 Assigning Fields to Graphs 32.10 Interpretation of the Theory of Graphs in the Theory of Fields Exercises Notes Chapter 33 Algebraically Closed Fields with Distinguished Automorphisms 33.1 The Base Field 33.2 Coding in PAC Fields with Monadic Quantifiers 33.3 The Theory of Almost all ⟨K, σ1, . . . , σe⟩’s 33.4 The Probability of Truth Sentences Chapter 34 Galois Stratification 34.1 The Artin Symbol 34.2 Conjugacy Domains under Projections 34.3 Normal Stratification 34.4 Elimination of One Variable 34.5 The Complete Elimination Procedure 34.6 Model-Theoretic Applications 34.7 A Limit of Theories Exercises Notes Chapter 35 Galois Stratification over Finite Fields 35.1 The Elementary Theory of Frobenius Fields 35.2 The Elementary Theory of Finite Fields 35.3 Near Rationality of the Zeta Function of a Galois Formula Exercises Notes Chapter 36 Problems of Field Arithmetic 36.1 Solved Problems 36.2 Open Problems References Index This book uses algebraic tools to study the elementary properties of classes of fields and related algorithmic problems. The first part covers foundational material on infinite Galois theory, profinite groups, algebraic function fields in one variable and plane curves. It provides complete and elementary proofs of the Chebotarev density theorem and the Riemann hypothesis for function fields, together with material on ultraproducts, decision procedures, the elementary theory of algebraically closed fields, undecidability and nonstandard model theory, including a nonstandard proof of Hilbert's irreducibility theorem. The focus then turns to the study of pseudo algebraically closed (PAC) fields, related structures and associated decidability and undecidability results. PAC fields (fields K with the property that every absolutely irreducible variety over K has a rational point) first arose in the elementary theory of finite fields and have deep connections with number theory. This fourth edition substantially extends, updates and clarifies the previous editions of this celebrated book, and includes a new chapter on Hilbertian subfields of Galois extensions. Almost every chapter concludes with a set of exercises and bibliographical notes. An appendix presents a selection of open research problems. Drawing from a wide literature at the interface of logic and arithmetic, this detailed and self-contained text can serve both as a textbook for graduate courses and as an invaluable reference for seasoned researchers. 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)?