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Field arithmetic

By: Contributor(s): Material type: TextTextLanguage: English Series: Ergebnisse der Mathematik und ihrer GrenzgebietePublication details: New York Springer-Verlag 1986Description: xvi, 458pISBN:
  • 3540166408 (HB)
Subject(s):
Contents:
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
Summary: 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.
Item type: BOOKS
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IMSc Library IMSc Library 511.22 FRI (Browse shelf(Opens below)) Available 22070

Includes index

Includes bibliography (p. 445-451) and references

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

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.

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The Institute of Mathematical Sciences, Chennai, India