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Series 
Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics, 00711136 ; 11


Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics ; 11

Note 
Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely selfcontained 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 finitefield 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 profree. 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 profree absolute Galois group (includes Shafarevich's conjecture)? 
Link 
Print version: 9783662072189

Subject 
Mathematics


Geometry, Algebraic


Field theory (Physics)


Logic, Symbolic and mathematical


Field theory (Physics) fast (OCoLC)fst00923918


Geometry, Algebraic. fast (OCoLC)fst00940902


Logic, Symbolic and mathematical. fast (OCoLC)fst01002068


Mathematics. fast (OCoLC)fst01012163


Electronic books

Alt Author 
Jarden, Moshe

