Purely inseparable field extension

An algebraic extension in characteristic $p>0$ is purely inseparable when every one of its elements is purely inseparable over the base field.

An algebraic element $x$ is purely inseparable over $K$ when $x^{p^n}\in K$ for some $n\ge 0$.

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 $p$.

If $M/L$ is purely inseparable, an embedding of $L$ into an algebraic closure has at most one extension to $M$: if $x^{p^n}\in L$, the image of $x$ must be the unique $p^n$th root of the image of $x^{p^n}$. Existence in a normal overfield therefore implies uniqueness.