Field Arithmetic

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)? Front Matter....Pages I-XVII Infinite Galois Theory and Profinite Groups....Pages 1-11 Algebraic Function Fields of One Variable....Pages 12-27 The Riemann Hypothesis for Function Fields....Pages 28-42 Plane Curves....Pages 43-53 The Čebotarev Density Theorem....Pages 54-73 Ultraproducts....Pages 74-87 Decision Procedures....Pages 88-100 Algebraically Closed Fields....Pages 101-108 Elements of Algebraic Geometry....Pages 109-128 Pseudo Algebraically Closed Fields....Pages 129-140 Hilbertian Fields....Pages 141-149 The Classical Hilbertian Fields....Pages 150-160 Nonstandard Structures....Pages 161-169 Nonstandard Approach to Hilbert’s Irreducibility Theorem....Pages 170-182 Profinite Groups and Hilbertian Fields....Pages 183-200 The Haar Measure....Pages 201-227 Effective Field Theory and Algebraic Geometry....Pages 228-247 The Elementary Theory of e -free PAC Fields....Pages 248-267 Examples and Applications....Pages 268-285 Projective Groups and Frattini Covers....Pages 286-313 Perfect PAC Fields of Bounded Corank....Pages 314-325 Undecidability....Pages 326-351 Frobenius Fields....Pages 352-367 On ω -free PAC Fields....Pages 368-402 Galois Stratification....Pages 403-421 Galois Stratification over Finite Fields....Pages 422-441 Back Matter....Pages 442-460