Suppose $A$ is a $k$-algebra, with $k$ a field, and let $\ell$ be a field extension of $k$. Is there an easy way to see/recover $\mathrm{Spec}(A)$ in/from $\mathrm{Spec}(A \otimes_k \ell)$, using the action of $\mathrm{Aut}(\ell/k)$ ? *Important special case*: $\ell$ is an algebraic closure of $k$.

To illustrate my question, let $\ell$ be an algebraic closure of $k$. Then a maximal ideal $\frak{m}$ of $A$ corresponds to an $\mathrm{Aut}(\ell/k)$-orbit of maximal ideals in $A \otimes_k \ell$ (the maximal ideals in $A \otimes_k \ell$ ''over $\frak{m}$''). Is such a thing true for the other prime ideals (and for any field extension $\ell$, not ''just'' algebraic closures) ? Is there a similar way (so using the $\mathrm{Aut}(\ell/k)$-action as above) to describe the closed sets of $\mathrm{Spec}(A)$ ?

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