  Let $F$ be any field. We give a short and elementary proof that any finite
subgroup $G$ of $PGL(2,F)$ occurs as a Galois group over the function field
$F(x)$. We also develop a theory of descent to subfields of $F$. This enables
us to realize the automorphism groups of finite subgroups of $PGL(2,F)$ as
Galois groups.
