  We study the relationship between the local and global Galois theory of
function fields over a complete discretely valued field. We give necessary and
sufficient conditions for local separable extensions to descend to global
extensions, and for the local absolute Galois group to inject into the global
absolute Galois group. As an application we obtain a local-global principle for
the index of a variety over such a function field. In this context we also
study algebraic versions of van Kampen's theorem, describing the global
absolute Galois group as a pushout of local absolute Galois groups.
