The Galois group of a stable homotopy theory

To a “stable homotopy theory” (a presentable, symmetric monoidal stable ∞-category), we naturally associate a category of finite étale algebra objects and, using Grothendieck's categorical machine, a profinite group that we call the Galois group. We then calculate the Galois groups in several examples. For instance, we show that the Galois group of the periodic E∞E∞-algebra of topological modular forms is trivial and that the Galois group of K(n)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. A fundamental idea throughout is the purely categorical notion of a “descendable” algebra object and an associated analog of faithfully flat descent in this context.