Field arithmetic /

Field arithmetic /

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Bibliographic Details
Author / Creator:Fried, Michael D., 1942-
Edition:3rd ed., rev. / by Moshe Jarden.
Imprint:Berlin : Springer, c2008.
Description:xxiii, 792 p. : ill. ; 24 cm.
Language:English
Series:Ergebnisse der Mathematik und ihrer Grenzgebiete, 0071-1136 ; 3. Folge, v. 11 = A series of modern surveys in mathematics
Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 11.
Subject:
Algebraic fields.
Algebraic number theory.
Algebraic fields.
Algebraic number theory.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/7186232
Hidden Bibliographic Details
Other authors / contributors:Jarden, Moshe, 1942-
ISBN:9783540772699 (hd. bd.)
3540772693 (hd. bd.)
Notes:Includes bibliographical references (p. [761]-779) and index.
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 Prüfer Group
  • 1.5. The Absolute Galois Group of a Finite Field
  • Exercises
  • Notes
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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 Fields
  • Exercises
  • Notes
  • Chapter 8. Decision Procedures
  • 8.1. Deduction Theory
  • 8.2. Gödel'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
  • Chapter 9. Algebraically Closed Fields
  • 9.1. Elimination of Quantifiers
  • 9.2. A Quantifiers Elimination Procedure
  • 9.3. Effectiveness
  • 9.4. Applications
  • Exercises
  • Notes
  • 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
  • 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 K ins 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
  • 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
  • 13.6. Non-Hilbertian g-Hilbertian Fields
  • 13.7. Twisted Wreath Products
  • 13.8. The Diamond Theorem
  • 13.9. Weissauer's Theorem
  • Exercises
  • Notes
  • 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
  • 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
  • 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 {{\cal Z}}/2^n{{\cal Z}} -extensions
  • 16.6. {{\cal Z}}_p -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
  • Chapter 17. Free Profinite Groups
  • 17.1. The Rank of a Profinite Group
  • 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
  • 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
  • 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
  • Chapter 20. The Elementary Theory of e-Free PAC Fields
  • 20.1. N 1 -Saturated PAC Fields
  • 20.2. The Elementary Equivalence Theorem of N 1 -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 K s (¿ 1 ,..., ¿ e )
  • 20.9. The Transfer Theorem
  • 20.10. The Elementary Theory of Finite Fields
  • Exercises
  • Notes
  • Chapter 21. Problems of Arithmetical Geometry
  • 21.1. The Decomposition-Intersection Procedure
  • 21.2. C i -Fields and Weakly C i -Fields
  • 21.3. Perfect PAC Fields which are C i
  • 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. Generalized Carlitz's Conjecture
  • Exercises
  • Notes
  • 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, {{\cal Z}}_p )
  • 22.14. The General Linear Group GL(2, {{\cal Z}}_p )
  • Exercises
  • Notes
  • 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
  • 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
  • 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
  • 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
  • 26.3. Random e-tuples in \hat {{\op Z}}^n
  • 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 Fields
  • 27.3. New Non-Classical Hilbertian Fields
  • 27.4. An abundance of ¿-Free PAC Fields
  • Notes
  • 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
  • Chapter 29. Algebraically Closed Fields with Distinguished Automorphisms
  • 29.1. The Base Field K
  • 29.2. Coding in PAC Fields with Monadic Quantifiers
  • 29.3. The Theory of Almost all ⟨ \tilde {{K}} , ¿ 1 , ..., ¿ e ⟩'s
  • 29.4. The Probability of Truth Sentences
  • 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
  • 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 Formula
  • Exercises
  • Notes
  • Chapter 32. Problems of Field Arithmetic
  • 32.1. Open Problems of the First Edition
  • 32.2. Open Problems of the Second Edition
  • 32.3. Open problems
  • References
  • Index
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