Field Arithmetic (ERGEBNISSE DER MATHEMATIK UND IHRER GRENZGEBIETE 3 FOLGE) - Hardcover

Fried, M.D.

Hardcover

ISBN 10: 0387166408 ISBN 13: 9780387166407

Publisher: Springer Nature, 1986

Synopsis

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)?

About this title

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 second edition:

"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)

"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)

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.

 

Edition details

• Publisher: Springer Nature
• Publication date: 1986
• Language: English
• ISBN 10: 0387166408
• ISBN 13: 9780387166407
• Binding: Hardcover