  We discuss some of the basic ideas of Galois theory for commutative
S-algebras originally formulated by John Rognes. We restrict attention to the
case of finite Galois groups and to global Galois extensions.
  We describe parts of the general framework developed by Rognes. Central roles
are played by the notion of strong duality and a trace mapping constructed by
Greenlees and May in the context of generalized Tate cohomology. We give some
examples where algebraic data on coefficient rings ensures strong topological
consequences. We consider the issue of passage from algebraic Galois extensions
to topological ones applying obstruction theories of Robinson and
Goerss-Hopkins to produce topological models for algebraic Galois extensions
and the necessary morphisms of commutative S-algebras. Examples such as the
complex K-theory spectrum as a KO-algebra indicate that more exotic phenomena
occur in the topological setting. We show how in certain cases topological
abelian Galois extensions are classified by the same Harrison groups as
algebraic ones and this leads to computable Harrison groups for such spectra.
We end by proving an analogue of Hilbert's theorem 90 for the units associated
with a Galois extension.
