Journal für die reine und angewandte Mathematik

J reine angew Math 406 (1990), 200—218 Journal für die reine und

angewandte Mathematik

© Walter de Gruyter Berlin New York 1990

On the Galois Theory of function fields of one variable over number fields

By Florian Pop at Heidelberg

Introduction and main results

For an arbitrary field К we denote by К the algebraic closure of К and by G|^ = Aut(K|K) the absolute Galois group of К endowed with the Krull topology.

The main result we prove in this paper is:

Theorem . Let F\Q, E\Q be two function fields of one variable {Q not necessarily the exact constant field of F or E). If Gp and Gp are isomorphic, then F and E are isomorphic.

More precisely, for each group isomorphism Ф : Gp ^^ Gp there exists a field isomorphism ф: E -^ F with the property:

and Ф is unique with the property above. As a consequence ф maps E isomorphically onto F.

As corollaries one obtains:

Theorem . Let F\Q and E\Q be as above. Denote by Homloc(Gp., Gp) the space of all group homomorphisms from Gp to Gp which are local homeomorphisms, and by Inn(G£) the group of all inner automorphisms of Gp. Then Inn(G£) acts in a canonical way on Homloc {Gp, Gp) and the canonical mapping

Hom ( E|F ) -^ Homloc(Gf., G^VInnCG^)

is a bijection.

Theorem . Let Q^ be the algebraically closed field of characteristic 0 and absolute transcendence degree 1. Let Aut(ßi) denote the automorphism group of Q^ endowed with the weak topology. Then any continuous automorphism of Aut{Qi) is inner.