The Galois group in stable homotopy theory

To a "stable homotopy theory," we naturally associate a
category of finite etale algebra objects and, using Grothendieck's
categorical machine, a profinite group that we call the Galois group.
The Galois group contains a purely algebraic piece coming from the
classical theory, but in general there is an additional (topological)
component. This was first observed by Rognes in the case of periodic
real K-theory.

We calculate the Galois groups in several examples. For instance, we
show that the Galois group of the periodic E_\infty-algebra of
topological modular forms is trivial and that the Galois group of
K(n)-local stable homotopy theory is an extended version of the Morava

stabilizer group. We also describe the Galois group of the stable
module category of a finite group, using a new technique of
"S^1-descent.”