Let's consider an arbitrary field $K$ and raise the following question: in which sense can we approximate $K$ by a perfect field? Any reasonable notion of approximation by a perfect field should admit $K$ as the best possible approximation in that sense if $K$ is already perfect. We know that if $char(K)=0$ then $K$ is perfect, so we may concentrate on the positive characteristic case.
The problem would be the following:
Given a non-perfect field $K$ of positive characteristic, in which sense can we approximate $K$ by a perfect field?
I'm aware that to attack a problem like this it should be desirable to have a goal in mind. That is, it would be good to define a notion of proximity or similarity that allows us to say "this field is a good approximation to that other field, good enough to solve this kind of problem".
Having said that, I'm going to consider, with no particular goal in mind, the following notion of best approximation by a perfect field from below:
Definition Given a field $K$, we say that a subfield $L\subset K$ is the best approximation of $K$ by a perfect field from below if $L$ is a perfect field and for every perfect subfield $F$ of $K$, we have $F\subset L$.
If $char(K)=0$ then $K$ itself is the best approximation of $K$ by a perfect field from below. If $char(K)=p>0$, it can be shown that $L=\bigcap_{n\geq 1} K^{p^n}$ is the best approximation of $K$ by a perfect field from below. Note that the use of the best approximation of $K$ by a perfect field from below is appropriate because we have uniqueness.
Now I'd like to define the dual notion of best approximation of $K$ by a perfect field from above. Two problems here. First, we must lift above the original field $K$ and there is no canonical way of doing that. This is not a big problem because if we define a satisfactory notion, two such approximations from above should be isomorphic as field extensions of $K$.
Definition Given a field $K$, we say that a field $L\supset K$ is a best approximation of $K$ by a perfect field from above if $L$ is a perfect field and for every perfect subfield $F$ of $L$ with $F\supset K$, we have $L\subset F$.
Againg, if $char(K)=0$ then $K$ itself is a best approximation of $K$ by a perfect field from above.
The second problem is that I don't even know if it's true that for any field $K$ with $char(K)=p>0$ there exist best approximations of $K$ by perfect fields from above. Let alone showing that two such approximations are isomorphic as field extensions of $K$.
My questions:
Are the last two claims true? If not, some counterexamples?
If they are true, any ideas on how to proceed for a proof?
Thanks!
Yes, there is a best perfect extension field $K^{p^{-\infty}}$ of any given field $K$ of characteristic $p$, called the perfect closure of $K$ (more about the article "the" below).
The field $K^{p^{-\infty}}$ consists of all elements $l\in \bar K$ in an algebraic closure $\bar K$ of $K$ with the property that $l^{p^r}\in K $ for some $r\geq 0$.
It is perfect and has the property you call "best approximation from above" with respect to perfect extensions of $K$.
The icing on the cake is that, although it is constructed from an algebraic closure $\bar K$ of $K$, it is more canonical than $\bar K$.
Indeed, if $L$ is any perfect extension of $K$ there is a unique $K$-morphism of extensions $L\to K$.
Hence perfect closures are unique up to unique isomorphism, whereas in contrast algebraic closures are unique up to non-unique isomorphism.