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.
|Original language||English GB|
|Title of host publication||Geometry & Topology Monographs 3|
|Publisher||Mathematical Sciences Publishers|
|Edition||Part II section 7|
|State||Published - 2000|
|Name||Geometry & Topology Monographs |
- 12E30, 12J25, 19M05