Let $K$ be a perfect field, and let $A$ be a unital, associative, commutative $K$-algebra. Let $K^{\operatorname{sep}}$ be the separable closure of of $K$, and $\Gamma = \operatorname{Gal}(K^{\operatorname{sep}}/K)$ be the absolute Galois group.
I want to define some sort of action of $\Gamma$ on $A$. If $A = L$ is a field, this is not problematic, since then $L/K$ is a separable extension, so $L$ embeds into $K^{\operatorname{sep}}$, and $\Gamma$ will (setwise) fix (the image of) $L$ inside $K^{\operatorname{sep}}$, so $\Gamma$ acts on $L$.
More generally, if $A$ is not a field, my intuition is that I should be able to do something like: take a maximal subfield $L \subset A$, and then view $A$ as an $L$-algebra, and just have $\Gamma$ act on $A$ by acting on the field of scalars $L$.
My main reservation/question is the following.
Does $A$ necessarily have a "unique" maximal subfield?
Here, "unique" should mean whatever it needs to mean for the action of $\Gamma$ on $A$ to be well defined, in the sense of not depending on the choice of maximal subfield, if there are multiple. My best guess for this is that such subfields should be conjugate in $A$, although perhaps a stronger notion of uniqueness is needed.
A related question is:
Should "maximal'' be with respect to inclusion, or with respect to dimension, or are they just the same thing?
I know that there are results from the theory of Brauer groups regarding maximal subfields of (possibly noncommutative) division algebras and and central simple algebras, but in my case I don't want to assume central simple or anything like that, but I am assuming my algebra is commutative.
What you can say is that $A$ has a maximal ind-étale subalgebra. Under some reasonable finiteness conditions on $A$ this will end up being genuinely étale, so it'll be some finite product of finite separable extensions of $k$. The Spec of this thing is a kind of "étale $\pi_0$" of $\text{Spec } A$.
There's a perfectly good Galois action on $A \otimes_k k_s$ already, though.