The author introduces the notion of a Galois extension of commutative \(S\)-algebras (\(E_\infty\) ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–Mac Lane spectra of commutative rings, real and complex topological \(K\)-theory, Lubin–Tate spectra and cochain \(S\)-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and étale extensions of commutative \(S\)-algebras, and the Goerss–Hopkins–Miller theory for \(E_\infty\) mapping spaces. He shows that the global sphere spectrum \(S\) is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava \(K\)-theories. He also defines Hopf–Galois extensions of commutative \(S\)-algebras and studies the complex cobordism spectrum \(MU\) as a common integral model for all of the local Lubin–Tate Galois extensions.

The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the \(p\)-complete study for \(p\)-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the \(E\)-local stable homotopy category, for any spectrum \(E\).