Frobenius factorization through a finite normal cover ($L^p\subseteq K\text{or}K\subseteq L^p$)

For a finite surjective map of normal integral schemes with function fields $K\subseteq L$ in characteristic $p$, the condition $L^p\subseteq K$ lets the Absolute Frobenius morphism of the source factor through the map. On an affine chart $A\subseteq B$, send $b$ to $b^p\in A$: it lies in $K$, is integral over $A$, and $A$ is integrally closed. The condition $K\subseteq L^p$ instead lets the map factor after the source Frobenius. Send $a$ to its unique $p$th root in $L$; that root lies in the normal ring $B$ because it satisfies a monic polynomial over $B$. These rules localize and glue; injectivity of Frobenius in the function field supplies uniqueness.