Field Arithmetic: (11 Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)

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 first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)? The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.

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.

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.

Review :
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)