The local correspondence over absolute fields: an algebraic approach (Q2708379)

Let \(K, K'\) be infinite fields which are finitely generated over their prime fields. Pop proved using model-theoretic methods that any isomorphism of the absolute Galois groups of \(K\) and \(K'\) maps the decomposition groups of the Zariski prime divisors on \(K\) bijectively onto the decomposition groups of the Zariski prime divisors on \(K'\) (relative to the separable closures). This was a main ingredient in his proof of the 0-dimensional case of Grothendieck's anabelian conjecture. In this paper we give a simplified and purely algebraic proof of this fact.