Geometry & Topology Monographs 3 (2000) -
Invitation to higher local fields,
Part II, section 7, pages 273-279
Recovering higher global and local fields from Galois groups -
an algebraic approach
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
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.
AMS subject classification. 12E30, 12J25, 19M05.
Department of Mathematics, Ben Gurion University of the Negev,
P.O.Box 653, Be'er-Sheva, 84105 Israel
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