I have the following proven theorem:
Let $L:K$ be a finite field extension and $\phi: K\xrightarrow{} L'$ a field homomorphism. Then there exist at most $[L:K]$ different field homomorphism $\widetilde{\phi}: L\xrightarrow{} L'$ with $\widetilde{\phi}\big|_{K}=\phi$.
Where $[L:K]$ denotes the dimension of $L$ over $K$.
Now my question is:
If I set $L=L'$ and $\phi =id$, the identity, will then $\widetilde{\phi}$ automatically become a field automorphism?
Thanks in advance!