Field Arithmetic

by Michael D. Fried, Moshe Jarden

Overview

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.

Thisfourth 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.

Product Details

ISBN-13:	9783031280191
Publisher:	Springer Nature Switzerland
Publication date:	06/13/2023
Series:	Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics , #11
Edition description:	Fourth Edition 2023
Pages:	827
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Moshe Jarden (revised and considerably enlarged the book in 2004 (2nd edition) and revised again in 2007 (the present 3rd edition).

Born on 23 August, 1942 in Tel Aviv, Israel.

Education:

Ph.D. 1969 at the Hebrew University of Jerusalem on

"Rational Points of Algebraic Varieties over Large Algebraic Fields".

Thesis advisor: H. Furstenberg.

Habilitation at Heidelberg University, 1972, on

"Model Theory Methods in the Theory of Fields".

Positions:

Dozent, Heidelberg University, 1973-1974.

Seniour Lecturer, Tel Aviv University, 1974-1978

Associate Professor, Tel Aviv University, 1978-1982

Professor, Tel Aviv University, 1982-

Incumbent of the Cissie and Aaron Beare Chair,

Tel Aviv University. 1998-

Academic and Professional Awards

Fellowship of Alexander von Humboldt-Stiftung in Heidelberg, 1971-1973.

Fellowship of Minerva Foundation, 1982.

Chairman of the Israel Mathematical Society, 1986-1988.

Member of the Institute for Advanced Study, Princeton, 1983, 1988.

Editor of the Israel Journal of Mathematics, 1992-.

Landau Prize for the book "Field Arithmetic". 1987.

Director of the Minkowski Center for Geometry founded by the

Minerva Foundation, 1997-.

L. Meitner-A.v.Humboldt Research Prize, 2001

Member, Max-Planck Institut f\"ur Mathematik in Bonn, 2001.

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.

Editorial Reviews

From the reviews of the second edition:

"This second and considerably enlarged edition reflects the progress made in field arithmetic during the past two decades. … The book also contains very useful introductions to the more general theories used later on … . the book contains many exercises and historical notes, as well as a comprehensive bibliography on the subject. Finally, there is an updated list of open research problems, and a discussion on the impressive progress made on the corresponding list of problems made in the first edition." (Ido Efrat, Mathematical Reviews, Issue 2005 k)

"The goal of this new edition is to enrich the book with an extensive account of the progress made in this field … . the book is a very rich survey of results in Field Arithmetic and could be very helpful for specialists. On the other hand, it also contains a large number of results of independent interest, and therefore it is highly recommendable to many others too." (Roberto Dvornicich, Zentralblatt MATH, Vol. 1055, 2005)

From the reviews of the third edition:

"The book give an introduction to the arithmetic of fields that is fairly standard, covering infinite Galois theory, profinite groups, extension of valued fields, algebraic function fields … and an introduction to affine and projective curves providing a geometric interpretation for results formulated in the language of function fields. … It could be used a text for graduate students entering the field, since the material is so well organized, even including exercises at the end of every chapter." (Felipe Zaldivar, MAA Online, December, 2008)

From the Publisher