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)?
CONTENT
1. Infinite Galois Theory and Profinite Groups -- 2. Algebraic Function Fields of One Variable -- 3. The Riemann Hypothesis for Function Fields -- 4. Plane Curves -- 5. The ?ebotarev Density Theorem -- 6. Ultraproducts -- 7. Decision Procedures -- 8. Algebraically Closed Fields -- 9. Elements of Algebraic Geometry -- 10. Pseudo Algebraically Closed Fields -- 11. Hilbertian Fields -- 12. The Classical Hilbertian Fields -- 13. Nonstandard Structures -- 14. Nonstandard Approach to Hilbertโ{128}{153}s Irreducibility Theorem -- 15. Profinite Groups and Hilbertian Fields -- 16. The Haar Measure -- 17. Effective Field Theory and Algebraic Geometry -- 18. The Elementary Theory of e-free PAC Fields -- 19. Examples and Applications -- 20. Projective Groups and Frattini Covers -- 21. Perfect PAC Fields of Bounded Corank -- 22. Undecidability -- 23. Frobenius Fields -- 24. On ?-free PAC Fields -- 25. Galois Stratification -- 26. Galois Stratification over Finite Fields -- Open Problems -- References
