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.
