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Catalogue Information
Field name
Details
Dewey Class
512.3
Title
Field Arithmetic ([EBook] /) / by Michael D. Fried, Moshe Jarden.
Author
Fried, Michael D.
Added Personal Name
Jarden, Moshe
author.
Other name(s)
SpringerLink (Online service)
Publication
Berlin, Heidelberg : : Springer Berlin Heidelberg : : Imprint: Springer, , 1986.
Physical Details
XVII, 460 p. 2 illus. : online resource.
Series
Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics
0071-1136 ; ; 11
ISBN
9783662072165
Summary Note
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)?:
Contents note
1. Infinite Galois Theory and Profinite Groups -- 2. Algebraic Function Fields of One Variable -- 3. The Riemann Hypothesis for Function Fields -- 4. Plane Curves -- 5. The ?ebotarev Density Theorem -- 6. Ultraproducts -- 7. Decision Procedures -- 8. Algebraically Closed Fields -- 9. Elements of Algebraic Geometry -- 10. Pseudo Algebraically Closed Fields -- 11. Hilbertian Fields -- 12. The Classical Hilbertian Fields -- 13. Nonstandard Structures -- 14. Nonstandard Approach to Hilbert’s Irreducibility Theorem -- 15. Profinite Groups and Hilbertian Fields -- 16. The Haar Measure -- 17. Effective Field Theory and Algebraic Geometry -- 18. The Elementary Theory of e-free PAC Fields -- 19. Examples and Applications -- 20. Projective Groups and Frattini Covers -- 21. Perfect PAC Fields of Bounded Corank -- 22. Undecidability -- 23. Frobenius Fields -- 24. On ?-free PAC Fields -- 25. Galois Stratification -- 26. Galois Stratification over Finite Fields -- Open Problems -- References.
System details note
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site
http://dx.doi.org/10.1007/978-3-662-07216-5
Links to Related Works
Subject References:
Algebra
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Algebraic Geometry
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Field Theory and Polynomials
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Field theory (Physics)
.
Mathematical logic
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Mathematical Logic and Foundations
.
Mathematics
.
Authors:
Fried, Michael D.
.
Jarden, Moshe
.
Corporate Authors:
SpringerLink (Online service)
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Series:
Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics
.
Classification:
512.3
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