Advertisement

Field Arithmetic

  • Michael D. Fried
  • Moshe Jarden

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
  21. Michael D. Fried, Moshe Jarden
    Pages 286-313
  22. Michael D. Fried, Moshe Jarden
    Pages 314-325
  23. Michael D. Fried, Moshe Jarden
    Pages 326-351
  24. Michael D. Fried, Moshe Jarden
    Pages 352-367
  25. Michael D. Fried, Moshe Jarden
    Pages 368-402
  26. Michael D. Fried, Moshe Jarden
    Pages 403-421
  27. Michael D. Fried, Moshe Jarden
    Pages 422-441
  28. Back Matter
    Pages 442-460

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

  • Michael D. Fried
    • 1
  • Moshe Jarden
    • 2
  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

  • DOI https://doi.org/10.1007/978-3-662-07216-5
  • Copyright Information Springer-Verlag Berlin Heidelberg 1986
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-662-07218-9
  • Online ISBN 978-3-662-07216-5
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site