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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)?

Algebraic fields. --- Algebraic number theory. --- Number theory --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra) --- Algebra. --- Geometry, algebraic. --- Field theory (Physics). --- Geometry. --- Logic, Symbolic and mathematical. --- Number theory. --- Algebraic Geometry. --- Field Theory and Polynomials. --- Mathematical Logic and Foundations. --- Number Theory. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Metamathematics --- Set theory --- Syllogism --- Euclid's Elements --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Algebraic geometry --- Geometry --- Mathematical analysis --- Number study --- Numbers, Theory of --- Algebra --- Algebraic geometry. --- Mathematical logic.

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This book uses algebraic tools to study the elementary properties of classes of fields and related algorithmic problems. The first part covers foundational material on infinite Galois theory, profinite groups, algebraic function fields in one variable and plane curves. It provides complete and elementary proofs of the Chebotarev density theorem and the Riemann hypothesis for function fields, together with material on ultraproducts, decision procedures, the elementary theory of algebraically closed fields, undecidability and nonstandard model theory, including a nonstandard proof of Hilbert's irreducibility theorem. The focus then turns to the study of pseudo algebraically closed (PAC) fields, related structures and associated decidability and undecidability results. PAC fields (fields K with the property that every absolutely irreducible variety over K has a rational point) first arose in the elementary theory of finite fields and have deep connections with number theory. This fourth edition substantially extends, updates and clarifies the previous editions of this celebrated book, and includes a new chapter on Hilbertian subfields of Galois extensions. Almost every chapter concludes with a set of exercises and bibliographical notes. An appendix presents a selection of open research problems. Drawing from a wide literature at the interface of logic and arithmetic, this detailed and self-contained text can serve both as a textbook for graduate courses and as an invaluable reference for seasoned researchers.

Algebra. --- Mathematics. --- Algebraic geometry. --- Algebraic fields. --- Polynomials. --- Geometry. --- Mathematical logic. --- Algebraic Geometry. --- Field Theory and Polynomials. --- Mathematical Logic and Foundations. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Euclid's Elements --- Algebra --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebraic number theory --- Rings (Algebra) --- Mathematical analysis --- Math --- Science --- Algebraic geometry --- Geometry --- Cossos algebraics --- Teoria algebraica de nombres