Extending a monomorphism $K \to E$ to an automorphism of $E$

Question

Suppose $E$ is a splitting field of a polynomial $f(x)$ over $F$ and $K$ a subfield of $E/F$. Can any monomorphism of $K \to E$ be extended to an automorphism of $E$?

Answer 1

Theorem 11.3 of Stewart, Galois Theory, 4th edition, says that if $L:K$ is a finite normal extension, and $M$ is an intermediate field, and $\tau$ is a $K$-monomorphism $M\to L$, then there is a $K$-automorphism $\sigma$ of $L$ such that $\sigma$ restricted to $M$ is $\tau$.

But Stewart is assuming here that $L$ is a subfield of the complex numbers and so, in particular, separable.