Field Arithmetic (4^th Ed., 4th ed. 2023)
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics Series, Vol. 11

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.

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.

Michael D. Fried received his PhD in Mathematics from the University of Michigan in 1967. After postdoctoral research at the Institute for Advanced Study (1967–1969), he became professor at Stony Brook University (8 years), the University of California at Irvine (26 years), the University of Florida (3 years) and the Hebrew University (2 years). He has held visiting positions at MIT, MSRI, the University of Michigan, the University of Florida, the Hebrew University, and Tel Aviv University. He has been an editor of several mathematics journals including the Research Announcements of the Bulletin of the American Mathematical Society and the Journal of Finite Fields and its Applications. His research is primarily in the geometry and arithmetic of families of nonsingular projective curve covers applied to classical moduli spaces using theta functions and l-adic representations. These are especially applied to relating the Regular Inverse Galois Problem and extensions of Serre's Open Image Theorem. He was included in 2013 Class of Fellows of the American Mathematical Society. He was also a Sloan Fellow (1972–1974), Lady Davis Fellow at Hebrew University (1987–1988), Fulbright scholar at Helsinki University (1982–1983), and Alexander von Humboldt Research Fellow (1994–1996).

Moshe Jarden received his PhD in Mathematics from the Hebrew University of Jerusalem in 1970 under the supervision of Hillel Furstenberg. His post-doctoral research was completed during the years 1971–1973 at the Institute of Mathematics, Heidelberg University, where he habilitated in 1972. In 1974, he returned to Israel, and joined the School of Mathematics of Tel Aviv University. He became a full professor in 1982, and the incumbent of the Cissie and Aaron Beare Chair in Algebra and Number Theory in 1998. His research focuses on families of large algebraic extensions of Hilbertian