Field Arithmetic

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Springer Science & Business Media, Apr 17, 2013 - Mathematics - 460 pages
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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)?

 

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Contents

Algebraic Function Fields of One Variable
12
The Riemann Hypothesis for Function Fields
28
Plane Curves
43
The Cebotarev Density Theorem
54
Ultraproducts
74
Decision Procedures
88
Algebraically Closed Fields
101
Elements of Algebraic Geometry
109
Effective Field Theory and Algebraic Geometry
228
The Elementary Theory of efree PAC Fields
248
Examples and Applications
268
Projective Groups and Frattini Covers
286
Perfect PAC Fields of Bounded Corank
314
Undecidability
326
Frobenius Fields
352
On apfree PAC Fields
368

Pseudo Algebraically Closed Fields
129
The Classical Hilbertian Fields
150
Nonstandard Approach to Hilberts Irreducibility Theorem
170
Profinite Groups and Hilbertian Fields
183
The Haar Measure
201
Characterization of Free Profinite Groups
375
Galois Stratification
403
Galois Stratification over Finite Fields
422
Open Problems
442
Copyright

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