Let $k$ be a field and $k^{\mathrm{sep}}$ one of its separable closures. Let $G_k$ be the absolute Galois group of $k$.
Now by abuse of notation, we define a group scheme on $\mathbf{\mathrm{Sch}}_k$ also by $G_k$: $$ G_k(U) = Top(|U|, G_k). $$
Let $G_k$ be the inverse limit of finite groups $G_i$, then this is the same as $\varprojlim_i \underline{G_i}^\#$ where $\#$ denotes sheafification with Zariski topology.
We can show that there is a right group action $\mathrm{Spec}(k^{\mathrm{sep}}) \times_k G_k \to \mathrm{Spec}(k^{\mathrm{sep}})$ by regarding $\mathrm{Spec}(k^{\mathrm{sep}})$ as inverse limit over $\mathrm{Spec}(K)$ where $K$ are finite galois extensions of $k$.
Now we can form the quotient stack $[\mathrm{Spec}(k^{\mathrm{sep}})/_k G_k]$. I know that this is isomorphic to $\mathrm{Spec}(k)$ when $[k^{\mathrm{sep}}: k] < \infty$. But what about the general case? How can we even show this is an algebraic stack?
Remark: It seems that the result depends on topology. To ensure $\mathrm{Spec}(k^{\mathrm{sep}})$ to be a $G_k$-torsor over $k$, it’s necessary to consider pro-etale topology.
