Minimal polynomial of a purely inseparable element

An element is purely inseparable over $K$ exactly when its minimal polynomial has the formTo 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.

Over $K={\mathbb{F}}_p(t)$, the polynomial $X^p-t$ is irreducible by the Eisenstein criterion in ${\mathbb{F}}_p[t][X]$. In any extension containing a root $\alpha$, characteristic $p$ givesIt therefore has exactly one distinct root, of multiplicity $p$.