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.
Keywords
field arithmetic, henselian valuations,
higher local fields