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.

Thisfourth 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.

Provides a self-contained account of the study of Diophantine fields through their absolute Galois groups Covers the prerequisites on infinite Galois theory, profinite groups, algebraic function fields and plane curves Gives a modern presentation of the theory of pseudo algebraically closed fields and related structures

Autorentext
Moshe Jarden (revised and considerably enlarged the book in 2004 (2nd edition) and revised again in 2007 (the present 3rd edition). Born on 23 August, 1942 in Tel Aviv, Israel. Education: Ph.D. 1969 at the Hebrew University of Jerusalem on "Rational Points of Algebraic Varieties over Large Algebraic Fields". Thesis advisor: H. Furstenberg. Habilitation at Heidelberg University, 1972, on "Model Theory Methods in the Theory of Fields". Positions: Dozent, Heidelberg University, 1973-1974. Seniour Lecturer, Tel Aviv University, 1974-1978 Associate Professor, Tel Aviv University, 1978-1982 Professor, Tel Aviv University, 1982- Incumbent of the Cissie and Aaron Beare Chair, Tel Aviv University. 1998- Academic and Professional Awards Fellowship of Alexander von Humboldt-Stiftung in Heidelberg, 1971-1973. Fellowship of Minerva Foundation, 1982. Chairman of the Israel Mathematical Society, 1986-1988. Member of the Institute for Advanced Study, Princeton, 1983, 1988. Editor of the Israel Journal of Mathematics, 1992-. Landau Prize for the book "Field Arithmetic". 1987. Director of the Minkowski Center for Geometry founded by the Minerva Foundation, 1997-. L. Meitner-A.v.Humboldt Research Prize, 2001 Member, Max-Planck Institut f"ur Mathematik in Bonn, 2001.

Inhalt

1 Infinite Galois Theory and Profinite Groups.- 2 Valuations.- 3 Linear Disjointness.- 4 Algebraic Function Fields of One Variable.- 5 The Riemann Hypothesis for Function Fields.- 6 Plane Curves.- 7 The Chebotarev Density Theorem.- 8 Ultraproducts.- 9 Decision Procedures.- 10 Algebraically Closed Fields.- 11 Elements of Algebraic Geometry.- 12 Pseudo Algebraically Closed Fields.- 13 Hilbertian Fields.- 14 The Classical Hilbertian Fields.- 15 The Diamond Theorem.- 16 Nonstandard Structures.- 17 The Nonstandard Approach to Hilbert's Irreducibility Theorem.- 18 Galois Groups over Hilbertian Fields.- 19 Small Profinite Groups.- 20 Free Profinite Groups.- 21 The Haar Measure.- 22 Effective Field Theory and Algebraic Geometry.- 23 The Elementary Theory of -Free PAC Fields.- 24 Problems of Arithmetical Geometry.- 25 Projective Groups and Frattini Covers.- 26 PAC Fields and Projective Absolute Galois Groups.- 27 Frobenius Fields.- 28 Free Profinite Groups of Infinite Rank.- 29 Random Elements in Profinite Groups.- 30 Omega-free PAC Fields.- 31 Hilbertian Subfields of Galois Extensions.- 32 Undecidability.- 33 Algebraically Closed Fields with Distinguished Automorphisms.- 34 Galois Stratification.- 35 Galois Stratification over Finite Fields.- 36 Problems of Field Arithmetic.