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

Let $A$ be an ${\mathbf{E}}_{\mathrm{\infty }}$-ring spectrum over the rational numbers. If $A$ satisfies a noetherian condition on its homotopy groups ${\pi }_{\ast }(A)$, we construct a collection of ${\mathbf{E}}_{\mathrm{\infty }}$-$A$-algebras that realize on homotopy the residue fields of ${\pi }_{\ast }(A)$. We prove an analog of the nilpotence theorem for these residue fields. As a result, we are able to give a complete algebraic description of the Galois theory of $A$ and of the thick subcategories of perfect $A$-modules. We also obtain partial information on the Picard group of $A$.