Realizability of algebraic Galois extensions by strictly commutative ring spectra

We discuss some of the basic ideas of Galois theory for commutative $\mathbb{S}$-algebras originally formulated by John Rognes. We restrict our attention to the case of finite Galois groups and to global Galois extensions.

We describe parts of the general framework developed by Rognes. Central rôles 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 by applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative $\mathbb{S}$-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.