In the problem statement, we are given $k \subset K$ an algebraic field extension of characteristic p > 0, L an algebraically closed field containing K, and $\delta: k \rightarrow L$ be an embedding. What I'm trying to show is that if there exists exactly one embedding $\tau: K \rightarrow L$ extending $\delta$ that $k \subset K$ is purely inseparable.
I've shown this when $k \subset K$ is finite, and now I'm trying to prove that I can reduce to this case.
Any hints would be much appreciated.