Lemma 9.15.10. Let $L/K$ be an algebraic normal extension of fields. Let $E/K$ be an extension of fields. Then either there is no $K$-embedding from $L$ to $E$ or there is one $\tau : L \to E$ and every other one is of the form $\tau \circ \sigma$ where $\sigma \in \text{Aut}(L/K)$.

Proof. Given $\tau$ replace $L$ by $\tau (L) \subset E$ and apply Lemma 9.15.7. $\square$

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