We introduce the notion of a Galois extension of commutative S-algebras (Emring spectra), often localized with respect to a fixed homology theory. There arenumerous examples, including some involving Eilenberg—Mac Lane spectra of com-mutative rings, real and complex topological K-theory, Lubin—Tate spectra andcochain S-algebras. We establish the main theorem of Galois theory in this gen-erality. Its proof involves the notions of separable and kale extensions of commu-tative S-algebras, and the Goerss—Hopkins—Miller theory for Em mapping spaces.We show that the global sphere spectrum S is separably closed, using Minkowski'sdiscriminant theorem, and we estimate the separable closure of its localization withrespect to each of the Morava K-theories. We also define Hopf—Galois extensionsof commutative S-algebras, and study the complex cobordism spectrum MU as acommon integral model for all of the local Lubin—Tate Galois extensions.
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