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Let $A$ be an $\mathbf{E}_{\infty}$-ring spectrum over the rational numbers.If $A$ satisfies a noetherian condition on its homotopy groups $\pi_*(A)$, weconstruct a collection of $\mathbf{E}_{\infty}$-$A$-algebras that realize onhomotopy the residue fields of $\pi_*(A)$. We prove an analog of the nilpotencetheorem for these residue fields. As a result, we are able to give a completealgebraic description of the Galois theory of $A$ and of the thicksubcategories of perfect $A$-modules. We also obtain partial information on thePicard group of $A$.

Residue fields for a class of rational $\mathbf{E}_\infty$-rings and applications