Field arithmetic / Michael D. Fried, Moshe Jarden.
Fried, Michael D., 1942- author.; Jarden, Moshe, 1942- author.
2023
QA247
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Title
Field arithmetic / Michael D. Fried, Moshe Jarden.
Author
Fried, Michael D., 1942- author.
Edition
Fourth edition.
ISBN
9783031280207 (electronic bk.)
3031280202 (electronic bk.)
9783031280191
3031280199
DOI
https://doi.org/10.1007/978-3-031-28020-7
Published
Cham : Springer, [2023]
Copyright
©2023
Language
English
Description
1 online resource (xxxi, 827 pages).
Item Number
10.1007/978-3-031-28020-7 doi
Call Number
QA247
Dewey Decimal Classification
512/.3
Summary
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.
Bibliography, etc. Note
Includes bibliographical references and index.
Access Note
Access limited to authorized users.
Source of Description
Online resource; title from PDF title page (SpringerLink, viewed June 21, 2023).
Added Author
Jarden, Moshe, 1942- author.
Series
Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 11. 2197-5655
Available in Other Form
Field Arithmetic
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1 Infinite Galois Theory and Profinite Groups
2 Valuations
3 Linear Disjointness
4 Algebraic Function Fields of One Variable
5 The Riemann Hypothesis for Function Fields
6 Plane Curves
7 The Chebotarev Density Theorem
8 Ultraproducts
9 Decision Procedures
10 Algebraically Closed Fields
11 Elements of Algebraic Geometry
12 Pseudo Algebraically Closed Fields
13 Hilbertian Fields
14 The Classical Hilbertian Fields
15 The Diamond Theorem
16 Nonstandard Structures
17 The Nonstandard Approach to Hilbert⁰́b9s Irreducibility Theorem
18 Galois Groups over Hilbertian Fields
19 Small Profinite Groups
20 Free Profinite Groups
21 The Haar Measure
22 Effective Field Theory and Algebraic Geometry
23 The Elementary Theory of ¿̐ư¿̐ư¿̐ư¿̐ư-Free PAC Fields
24 Problems of Arithmetical Geometry
25 Projective Groups and Frattini Covers
26 PAC Fields and Projective Absolute Galois Groups
27 Frobenius Fields
28 Free Profinite Groups of Infinite Rank
29 Random Elements in Profinite Groups
30 Omega-free PAC Fields
31 Hilbertian Subfields of Galois Extensions
32 Undecidability
33 Algebraically Closed Fields with Distinguished Automorphisms
34 Galois Stratification
35 Galois Stratification over Finite Fields
36 Problems of Field Arithmetic.

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