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In my Galois Theory notes we have the following theorem:

Let $L/K$ be a finite extension. TFAE:

(ii) For any extension $M/L$ and any $K$-homomorphism $\sigma:L \rightarrow M$, we have $\sigma(L)=L$.

(iii) $L/K$ is normal.

The proof of (ii)$\Rightarrow $(iii) reads

Suppose (ii) holds. Given any $\alpha \in L$, let $M/L$ be such that $f_\alpha^K$ splits in $M$, i.e. $f_\alpha^K=\prod (X-\alpha_i)$ for some $\alpha_i \in M$. We need to show $\alpha_i \in L$.

Take a $K$-homomorphism $\hat{\sigma}: K(\alpha) \rightarrow M$, $\alpha \mapsto \alpha_i$. Extend to a $K$-homomorphism $\sigma:L \rightarrow M$.

Why can we make this extension of $\hat{\sigma}$?

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