Perfect closure

Remarks

The set $K^{p^{-\infty}}$ is a subfield of $\overline{K}$, because $(\alpha + \beta)^{p^n} = \alpha^{p^n} + \beta^{p^n}$ in characteristic $p$. It is a perfect field, it is purely inseparable over $K$, and it is contained in every perfect subfield of $\overline{K}$ that contains $K$. It is unique up to a unique isomorphism over $K$.

Example: for $K = \F_p(t)$, the perfect closure is $\bigcup_{n \ge 0} \F_p(t^{1/p^n})$. Check: each $\F_p(t^{1/p^n})$ is purely inseparable over $K$, and the union is perfect because every element of $\F_p(t^{1/p^n})$ has a $p$-th root in $\F_p(t^{1/p^{n+1}})$.

Non-example: for $p$ odd, the field $\overline{K}$ is perfect and contains $K$, but it is not the perfect closure. It contains $\sqrt{t}$, and no power $\sqrt{t}^{\,p^n}$ lies in $K$, since $p^n$ is odd.