Field Arithmetic

by Fried, Michael D.; Jarden, Moshe

• ISBN13: 9783540228110
• ISBN10: 354022811X
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-01-01
• Publisher: Springer Verlag

Summary

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)?

Table of Contents

Infinite Galois theory and profinite groups	p. 1
Valuations and linear disjointness	p. 19
Algebraic function fields of one variable	p. 52
The Riemann hypothesis for function fields	p. 77
Plane curves	p. 95
The Chebotarev density theorem	p. 107
Ultraproducts	p. 132
Decision procedures	p. 149
Algebraically closed fields	p. 163
Elements of algebraic geometry	p. 172
Pseudo algebraically closed fields	p. 192
Hilbertian fields	p. 218
The classical Hilbertian fields	p. 230
Nonstandard structures	p. 266
Nonstandard approach to Hilbert's irreducibility theorem	p. 276
Galois groups over Hilbertian fields	p. 290
Free profinite groups	p. 337
The Haar measure	p. 362
Effective fields theory and algebraic geometry	p. 401
The elementary theory of e-free PAC fields	p. 427
Problems of arithmetical geometry	p. 452
Projective groups and Frattini covers	p. 494
PAC fields and projective absolute Galois groups	p. 541
Frobenius fields	p. 559
Free profinite groups of infinite rank	p. 591
Random elements in free profinite groups	p. 632
Omega-free PAC fields	p. 652
Undecidability	p. 668
Algebraically closed fields with distinguished automorphisms	p. 695
Galois stratification	p. 705
Galois stratification over finite fields	p. 727
Problems of field arithmetic	p. 748
Table of Contents provided by Blackwell. All Rights Reserved.