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© 1986

Field Arithmetic

Benefits

  • Second revised and substantially enlarged edition of the classical Ergebnisse Vol. 11 "Field Arithmetic", published in 1986

  • The second edition takes into account all the important new developments in the area of "Field Arithmetic"

  • The new edition gives full account of some of the unsolved problems, exposes others and adds new exciting topics

  • New topics include: Haran's diamond theorem, g-Hilbertian fields, GAR-realizations of finite simple groups, Schur's Conjecture, the Carlitz-Wan Conjecture, Melnikow formations

Book

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 11)

Table of contents

  1. Front Matter
    Pages I-XVII
  2. Michael D. Fried, Moshe Jarden
    Pages 1-11
  3. Michael D. Fried, Moshe Jarden
    Pages 12-27
  4. Michael D. Fried, Moshe Jarden
    Pages 28-42
  5. Michael D. Fried, Moshe Jarden
    Pages 43-53
  6. Michael D. Fried, Moshe Jarden
    Pages 54-73
  7. Michael D. Fried, Moshe Jarden
    Pages 74-87
  8. Michael D. Fried, Moshe Jarden
    Pages 88-100
  9. Michael D. Fried, Moshe Jarden
    Pages 101-108
  10. Michael D. Fried, Moshe Jarden
    Pages 109-128
  11. Michael D. Fried, Moshe Jarden
    Pages 129-140
  12. Michael D. Fried, Moshe Jarden
    Pages 141-149
  13. Michael D. Fried, Moshe Jarden
    Pages 150-160
  14. Michael D. Fried, Moshe Jarden
    Pages 161-169
  15. Michael D. Fried, Moshe Jarden
    Pages 170-182
  16. Michael D. Fried, Moshe Jarden
    Pages 183-200
  17. Michael D. Fried, Moshe Jarden
    Pages 201-227
  18. Michael D. Fried, Moshe Jarden
    Pages 228-247
  19. Michael D. Fried, Moshe Jarden
    Pages 248-267
  20. Michael D. Fried, Moshe Jarden
    Pages 268-285

About this book

Introduction

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)?

Keywords

Absolute Galois Groups Galois Stratification Galois group Galois theory Grad Hilbertian Fields Irreducibility PAC Fields Profinite Groups algebra algebraic geometry finite group ultraproduct

Authors and affiliations

  1. 1.Mathematical DepartmentUniversity of FloridaGainsvilleUSA
  2. 2.School of Mathematical Sciences, Raymond and Beverly Sackler, Faculty of Exact SciencesTel Aviv UniversityRamat Aviv, Tel AvivIsrael

Bibliographic information

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)