Let $K$ be a field of characteristic $p >0$ and $L/K$ an algebraic extension. We can assoicate to these fields two canonical operations:
The perfect closure a $K^{1/p^{\infty}} := \bigcup_{i \le 1} K^{1/p^i}$ and the other thing (I don't know it's name) $K(L^{p^\infty}) := \bigcap_{i \ge 1} K(L^{p^i})$.
Question: how are these operators related to each other in the sense of what we do know generally about intermediate fields $K((K^{1/p^\infty} \cap L)^{p^\infty})$ and $K(K(L^{p^\infty}))^{1/p^\infty} \cap L $ of $L/K$ which are as compositions of these two operations 'within the $L/K$ universe'?