  A main problem in Galois theory is to characterize the fields with a given
absolute Galois group. We apply a K-theoretic method for constructing
valuations to study this problem in various situations. As a first application
we obtain an algebraic proof of the 0-dimensional case of Grothendieck's
anabelian conjecture (proven by Pop), which says that finitely generated
infinite fields are determined up to purely inseparable extensions by their
absolute Galois groups. As a second application (which is a joint work with
Fesenko) we analyze the arithmetic structure of fields with the same absolute
Galois group as a higher-dimensional local field.
