Field Arithmetic

• Main Authors: Fried, Michael D. (Author), Jarden, Moshe (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg, 2005.
• Edition: Second Edition.
• Series: A Series of Modern Surveys in Mathematics ; 11
• Subjects: Mathematics.; Algebra.; Algebraic geometry.; Field theory (Physics).; Geometry.; Mathematical logic.; Number theory.; Algebraic Geometry.; Field Theory and Polynomials.; Mathematical Logic and Foundations.; Number Theory.

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 mea...

Table of Contents:

• Infinite Galois Theory and Profinite Groups
• Valuations and Linear Disjointness
• Algebraic Function Fields of One Variable
• The Riemann Hypothesis for Function Fields
• Plane Curves
• The Chebotarev Density Theorem
• Ultraproducts
• Decision Procedures
• Algebraically Closed Fields
• Elements of Algebraic Geometry
• Pseudo Algebraically Closed Fields
• Hilbertian Fields
• The Classical Hilbertian Fields
• Nonstandard Structures
• Nonstandard Approach to Hilbert’s Irreducibility Theorem
• Galois Groups over Hilbertian Fields
• Free Profinite Groups
• The Haar Measure
• Effective Field Theory and Algebraic Geometry
• The Elementary Theory of e-Free PAC Fields
• Problems of Arithmetical Geometry
• Projective Groups and Frattini Covers
• PAC Fields and Projective Absolute Galois Groups
• Frobenius Fields
• Free Profinite Groups of Infinite Rank
• Random Elements in Profinite Groups
• Omega-free PAC Fields
• Undecidability
• Algebraically Closed Fields with Distinguished Automorphisms
• Galois Stratification
• Galois Stratification over Finite Fields
• Problems of Field Arithmetic.