The absolute Galois group of a field 
is the group of field automorphisms
of a separable closure 
fixing 
:
 |
Its Krull topology is the inverse-limit topology from the finite Galois extensions inside 
,
so it is a profinite group. Using an algebraic
closure gives the same automorphism group, since
purely inseparable elements have unique conjugates. Different choices of separable
closure give isomorphic groups, with the
isomorphism defined only up to inner
automorphism.
Galois representations are representations of this group. The Bogomolov-Pop conjecture asks for reconstruction of certain function fields from much smaller quotients.
