Field arithmetic.(3rd ed., rev.)

Abstract

Infinite Galois Theory and profinite groups -- Valuations and linear disjointness -- Algebraic function fields of one variable -- The Riemann hypothesis for function fields -- Plane curves -- The Chebotarev density theorem -- Ultraproducts -- Decision procedures -- Algebraically closed fields -- Elements of algebraic geometry -- Pseudo algebraically closed fields -- Hilbertian fields -- The classical Hilbertian fields -- Nonstandard structures -- Nonstandard approach to Hilbert's irreducibility theorm -- Galois groups over Hilbertian fields -- Free profinite groups -- The Haar measure -- Effective field theory and algebraic geometry -- The elementary theory of e-Free PAC fields -- Problems of arithmetical geometry -- Projective groups and Frattini covers -- PAC fields and projective absolute Galois groups -- Frobenius fields -- Free profinite groups of infinite rank -- Random elements in free profinite groups -- Omega-free PAC fields -- Undecidability -- Algebraically closed fields with distinguished automorphisms -- Galois satisfaction -- Galois stratification over finite fields -- Problems of field arithmetic