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.
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.
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".
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 ContentsInfinite 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 Free Profinite Groups. - Omega-Free PAC Fields. - Undecidability. - Algebraically Closed Fields with Distinguished Automorphisms. - Galois Stratification. - Galois Stratification over Finite Fields. - Problems of Finite Arithmetic.