Field Arithmetic

Michael D. Fried, Moshe Jarden (auth.)3540772693, 9783540772699, 354022811X, 9783540228110

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)?

The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.

Table of contents :
Front Matter….Pages i-xxiv
Infinite Galois Theory and Profinite Groups….Pages 1-18
Valuations and Linear Disjointness….Pages 19-51
Algebraic Function Fields of One Variable….Pages 52-76
The Riemann Hypothesis for Function Fields….Pages 77-94
Plane Curves….Pages 95-106
The Chebotarev Density Theorem….Pages 107-131
Ultraproducts….Pages 132-148
Decision Procedures….Pages 149-162
Algebraically Closed Fields….Pages 163-171
Elements of Algebraic Geometry….Pages 172-191
Pseudo Algebraically Closed Fields….Pages 192-218
Hilbertian Fields….Pages 219-230
The Classical Hilbertian Fields….Pages 231-266
Nonstandard Structures….Pages 267-276
Nonstandard Approach to Hilbert’s Irreducibility Theorem….Pages 277-290
Galois Groups over Hilbertian Fields….Pages 291-337
Free Profinite Groups….Pages 338-362
The Haar Measure….Pages 363-402
Effective Field Theory and Algebraic Geometry….Pages 403-428
The Elementary Theory of e -Free PAC Fields….Pages 429-453
Problems of Arithmetical Geometry….Pages 454-496
Projective Groups and Frattini Covers….Pages 497-543
PAC Fields and Projective Absolute Galois Groups….Pages 544-561
Frobenius Fields….Pages 562-593
Free Profinite Groups of Infinite Rank….Pages 594-634
Random Elements in Profinite Groups….Pages 635-654
Omega-Free PAC Fields….Pages 655-670
Undecidability….Pages 671-697
Algebraically Closed Fields with Distinguished Automorphisms….Pages 698-707
Galois Stratification….Pages 708-729
Galois Stratification over Finite Fields….Pages 730-750
Problems of Field Arithmetic….Pages 751-760
Back Matter….Pages 761-792