Field Arithmetic

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 ?-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 ? with ?ins 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 ?-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/2?Z-Extensions
18.6 Z?-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 ?-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 ?-Free PAC Fields
23.5 The Elementary Theory of Perfect ?-Free PAC Fields
23.6 The Probable Truth of a Sentence
23.7 Change of Base Field
23.8 The Fields ?sep(?1, . . . , ??)
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 ??-Fields and Weakly ??-Fields
24.3 Perfect PAC Fields which are ??
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- ?-Groups
25.9 Supernatural Numbers
25.10 The Sylow Theorems
25.11 On Complements of Normal Subgroups
25.12 The Universal Frattini ?-Cover
25.13 Examples of Universal Frattini ?-covers
25.14 The Special Linear Group SL(2, Z?)
25.15 The General Linear Group GL(2, Z?)
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 ?-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-? Groups
29.3 Random Ẑ?
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 ?-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 ⟨?, ?1, . . . , ??⟩’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