Field Arithmetic

  • Michael D. Fried
  • Moshe Jarden

Part of the A Series of Modern Surveys in Mathematics book series (MATHE3, volume 11)

Table of contents

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

Arithmetic Counting Finite Galois theory Grad Irreducibility algebra algebraic geometry finite field function geometry number theory theorem ultraproduct variable

Authors and affiliations

  • Michael D. Fried
    • 1
  • Moshe Jarden
    • 2
  1. 1.Department of MathematicsMontana State University-BillingsBillingsUSA
  2. 2.School of MathematicsTel Aviv UniversityRamat Aviv Tel AvivIsrael

Bibliographic information

  • DOI https://doi.org/10.1007/b138352
  • Copyright Information Springer-Verlag Berlin Heidelberg 2005
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-22811-0
  • Online ISBN 978-3-540-26949-6
  • About this book