Minimal polynomial of a purely inseparable element
= Minimal polynomial of a purely inseparable element
An element is purely inseparable over $K$ exactly when its minimal polynomial has the form
$$
T^{p^n}-a,
\qquad a\in K.
$$
To prove the nontrivial direction, repeatedly factor a zero formal derivative through $T\mapsto T^p$ until the remaining irreducible polynomial has nonzero derivative. Since $T^{p^r}-x^{p^r}$ has only the root $x$, that remaining separable polynomial must be linear.