Suppose we have an irreducible polynomial $f\in K[x]$. Is there some way to sometimes tell whether $f$ splits completely after adjoining just one root of $f$ to $K$?

I am mostly interested in the case where $K$ is a function field $F_{q}(t_{1},\ldots,t_{m})$ \mathbb{F}_{q}(t_{1},\ldots,t_{m})$ over some finite field, so it might not be feasible to explicitly compute roots.

Suppose we have an irreducible polynomial $f\in K[x]$. Is there some way to sometimes tell whether $f$ splits completely after adjoining just one root of $f$ to $K$?

I am mostly interested in the case where $K$ is a function field $F_{q}(t_{1},\ldots,t_{m})$ over some finite field, so it might not be feasible to explicitly compute roots.