Ebook: Field Arithmetic
- Tags: Algebra, Algebraic Geometry, Field Theory and Polynomials, Geometry, Mathematical Logic and Foundations, Number Theory
- Series: A Series of Modern Surveys in Mathematics 11
- Year: 2005
- Publisher: Springer Berlin Heidelberg
- Language: English
- pdf
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)?
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)?
Content:
Front Matter....Pages i-xxiii
Infinite Galois Theory and Profinite Groups....Pages 1-18
Valuations and Linear Disjointness....Pages 19-51
Algebraic Function Fields of One Variable....Pages 52-76
The Riemann Hypothesis for Function Fields....Pages 77-94
Plane Curves....Pages 95-106
The Chebotarev Density Theorem....Pages 107-131
Ultraproducts....Pages 132-148
Decision Procedures....Pages 149-162
Algebraically Closed Fields....Pages 163-171
Elements of Algebraic Geometry....Pages 172-191
Pseudo Algebraically Closed Fields....Pages 192-217
Hilbertian Fields....Pages 218-229
The Classical Hilbertian Fields....Pages 230-265
Nonstandard Structures....Pages 266-275
Nonstandard Approach to Hilbert’s Irreducibility Theorem....Pages 276-289
Galois Groups over Hilbertian Fields....Pages 290-336
Free Profinite Groups....Pages 337-361
The Haar Measure....Pages 362-400
Effective Field Theory and Algebraic Geometry....Pages 401-426
The Elementary Theory of e-Free PAC Fields....Pages 427-451
Problems of Arithmetical Geometry....Pages 452-493
Projective Groups and Frattini Covers....Pages 494-540
PAC Fields and Projective Absolute Galois Groups....Pages 541-558
Frobenius Fields....Pages 559-590
Free Profinite Groups of Infinite Rank....Pages 591-631
Random Elements in Profinite Groups....Pages 632-651
Omega-free PAC Fields....Pages 652-667
Undecidability....Pages 668-694
Algebraically Closed Fields with Distinguished Automorphisms....Pages 695-704
Galois Stratification....Pages 705-726
Galois Stratification over Finite Fields....Pages 727-747
Problems of Field Arithmetic....Pages 748-754
Back Matter....Pages 755-780
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)?
Content:
Front Matter....Pages i-xxiii
Infinite Galois Theory and Profinite Groups....Pages 1-18
Valuations and Linear Disjointness....Pages 19-51
Algebraic Function Fields of One Variable....Pages 52-76
The Riemann Hypothesis for Function Fields....Pages 77-94
Plane Curves....Pages 95-106
The Chebotarev Density Theorem....Pages 107-131
Ultraproducts....Pages 132-148
Decision Procedures....Pages 149-162
Algebraically Closed Fields....Pages 163-171
Elements of Algebraic Geometry....Pages 172-191
Pseudo Algebraically Closed Fields....Pages 192-217
Hilbertian Fields....Pages 218-229
The Classical Hilbertian Fields....Pages 230-265
Nonstandard Structures....Pages 266-275
Nonstandard Approach to Hilbert’s Irreducibility Theorem....Pages 276-289
Galois Groups over Hilbertian Fields....Pages 290-336
Free Profinite Groups....Pages 337-361
The Haar Measure....Pages 362-400
Effective Field Theory and Algebraic Geometry....Pages 401-426
The Elementary Theory of e-Free PAC Fields....Pages 427-451
Problems of Arithmetical Geometry....Pages 452-493
Projective Groups and Frattini Covers....Pages 494-540
PAC Fields and Projective Absolute Galois Groups....Pages 541-558
Frobenius Fields....Pages 559-590
Free Profinite Groups of Infinite Rank....Pages 591-631
Random Elements in Profinite Groups....Pages 632-651
Omega-free PAC Fields....Pages 652-667
Undecidability....Pages 668-694
Algebraically Closed Fields with Distinguished Automorphisms....Pages 695-704
Galois Stratification....Pages 705-726
Galois Stratification over Finite Fields....Pages 727-747
Problems of Field Arithmetic....Pages 748-754
Back Matter....Pages 755-780
....