Overview

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.

Full Product Details

Author:   Michael D. Fried ,  Moshe Jarden
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   3rd ed. 2008
Volume:   11
Dimensions:   Width: 15.60cm , Height: 4.20cm , Length: 23.40cm
Weight:   2.900kg
ISBN:  

9783540772699

ISBN 10:   3540772693
Pages:   792
Publication Date:   06 May 2008
Audience:   Professional and scholarly ,  Professional & Vocational
Replaced By:   9783031280191
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Reviews

From the reviews of the second edition: <p> This second and considerably enlarged edition reflects the progress made in field arithmetic during the past two decades. a ] The book also contains very useful introductions to the more general theories used later on a ] . 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) <p> The goal of this new edition is to enrich the book with an extensive account of the progress made in this field a ] . 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)

Author Information

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.

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