 Unique embedding extension through a purely inseparable extension

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.