= Frobenius factorization through a finite normal cover
{c}
{title2=$L^p\subseteq K\quad\text{or}\quad K\subseteq L^p$}
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.
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