Title Field arithmetic / Michael D. Fried, Moshe Jarden

Edition 2nd ed., rev. and enl. / by Moshe Jarden Published Berlin ; New York : Springer, ©2005

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Description 1 online resource (xxii, 780 pages) : illustrations 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 Contents Cover -- Table of Contents -- Introduction to the Second Edition -- Introduction to the First Edition -- Notation and Convention -- 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 Prèufer 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 Extension 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 Theoren -- 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 Gèodel'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 Summary 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 fi Analysis wiskunde mathematics meetkunde geometry algebra algebraic geometry veeltermen polynomials getallenleer number theory Mathematics (General) Wiskunde (algemeen) Bibliography Includes bibliographical references (pages 755-768) and index Notes Print version record In OhioLINK electronic book center SpringerLink Subject Algebraic fields. Algebraic number theory. Corps algébriques. Nombres algébriques, théorie des. Teoria dos números. Álgebra. Corps algébrique. Théorie des nombres algébriques. Algebraic fields. Algebraic number theory. Cuerpos algebraicos Teoría algebraica de números Algebraic fields Algebraic number theory Teoria dos números. Álgebra. Corps algébrique. Théorie des nombres algébriques. Form Electronic book Author Jarden, Moshe, 1942- ISBN 9783540269496 3540269495 354022811X 9783540228110 6610305234 9786610305230

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