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.
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