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Field Arithmetic
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Catalogue Information
Field name
Details
Dewey Class
512
Title
Field Arithmetic ([Ebook]) / by Michael D. Fried, Moshe Jarden.
Author
Fried, Michael D.
Added Personal Name
Jarden, Moshe
Other name(s)
SpringerLink (Online service)
Edition statement
Third Edition.
Publication
Berlin, Heidelberg : Springer , 2008.
Physical Details
XXIV, 792 pages : online resource.
Series
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
0071-1136 ; ; 11
ISBN
9783540772705
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)? The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.:
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/10.1007/978-3-540-77270-5
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)
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Geometry
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Geometry, Algebraic
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Logic, Symbolic and mathematical
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Mathematical Logic and Foundations
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Number Theory
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Authors:
author
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Fried, Michael D.
.
Jarden, Moshe
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Corporate Authors:
SpringerLink (Online service)
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Series:
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
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Classification:
512
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