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We relate two different proposals to extend the \'etale topology intohomotopy theory, namely via the notion of finite cover introduced by Mathew andvia the notion of separable commutative algebra introduced by Balmer. We showthat finite covers are precisely those separable commutative algebras withunderlying dualizable module, which have a locally constant and finite degreefunction. We then use Galois theory to classify separable commutative algebrasin numerous categories of interest. Examples include the category of modulesover an $\mathbb{E}_\infty$-ring $R$ which is either connective or evenperiodic with $\pi_0(R)$ regular Noetherian, the stable module category of afinite group of $p$-rank one and the derived category of a qcqs scheme.

Separable commutative algebras and Galois theory in stable homotopy theories