LEADER 05414nam 22008175 450 001 9910731456903321 005 20240509140325.0 010 $a3-031-28020-2 024 7 $a10.1007/978-3-031-28020-7 035 $a(MiAaPQ)EBC30592746 035 $a(Au-PeEL)EBL30592746 035 $a(DE-He213)978-3-031-28020-7 035 $a(CKB)26938277700041 035 $a(EXLCZ)9926938277700041 100 $a20230612d2023 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aField Arithmetic /$fby Michael D. Fried, Moshe Jarden 205 $a4th ed. 2023. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2023. 215 $a1 online resource (839 pages) 225 1 $aErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,$x2197-5655 ;$v11 311 08$aPrint version: Fried, Michael D. Field Arithmetic Cham : Springer,c2023 9783031280191 327 $a1 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. 330 $aThis book uses algebraic tools to study the elementary properties of classes of fields and related algorithmic problems. The first part covers foundational material on infinite Galois theory, profinite groups, algebraic function fields in one variable and plane curves. It provides complete and elementary proofs of the Chebotarev density theorem and the Riemann hypothesis for function fields, together with material on ultraproducts, decision procedures, the elementary theory of algebraically closed fields, undecidability and nonstandard model theory, including a nonstandard proof of Hilbert's irreducibility theorem. The focus then turns to the study of pseudo algebraically closed (PAC) fields, related structures and associated decidability and undecidability results. PAC fields (fields K with the property that every absolutely irreducible variety over K has a rational point) first arose in the elementary theory of finite fields and have deep connections with number theory. This fourth edition substantially extends, updates and clarifies the previous editions of this celebrated book, and includes a new chapter on Hilbertian subfields of Galois extensions. Almost every chapter concludes with a set of exercises and bibliographical notes. An appendix presents a selection of open research problems. Drawing from a wide literature at the interface of logic and arithmetic, this detailed and self-contained text can serve both as a textbook for graduate courses and as an invaluable reference for seasoned researchers. 410 0$aErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,$x2197-5655 ;$v11 606 $aAlgebra 606 $aMathematics 606 $aAlgebraic geometry 606 $aAlgebraic fields 606 $aPolynomials 606 $aGeometry 606 $aMathematical logic 606 $aAlgebra 606 $aMathematics 606 $aAlgebraic Geometry 606 $aField Theory and Polynomials 606 $aGeometry 606 $aMathematical Logic and Foundations 606 $aCossos algebraics$2thub 606 $aTeoria algebraica de nombres$2thub 608 $aLlibres electrònics$2thub 615 0$aAlgebra. 615 0$aMathematics. 615 0$aAlgebraic geometry. 615 0$aAlgebraic fields. 615 0$aPolynomials. 615 0$aGeometry. 615 0$aMathematical logic. 615 14$aAlgebra. 615 24$aMathematics. 615 24$aAlgebraic Geometry. 615 24$aField Theory and Polynomials. 615 24$aGeometry. 615 24$aMathematical Logic and Foundations. 615 7$aCossos algebraics 615 7$aTeoria algebraica de nombres 676 $a658.40301 676 $a512.3 700 $aFried$b Michael D$055735 701 $aJarden$b Moshe$055736 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910731456903321 996 $aField arithmetic$9717341 997 $aUNINA