$\mathbf {The \ Problem \ is}:$ Let $E$ be the splitting field of a polynomial $f$ over $k.$ Let $k \subset K \subset E,$ then show that any $k-$monomorphism $\phi$ from $K \to E$ can be extended to a $k-$ automorphism from $E \to E.$
$\mathbf {My \ approach}:$ Let $[E : k]=n$ and let $B =\{\beta_i \}_{i=1}^n$ is a basis of $E$ over $k.$
Then, wlg, let $\{\beta_i \}_{i=1}^m$ be a basis of $K$ over $k.$ Then, $\beta_1=\{\phi (\beta_i) \}_{i=1}^m$ is a basis of some sub field of $E.$
Now, I am thinking we have to extend the basis $\beta_1$ of $K$ to that we obtain a basis of $E.$ is $k.$
A small hint is warmly appreciated . Thanks in advance .