Lemma 9.13.3. Let $K/F$ and $L/F$ be field extensions. Let $\sigma _1, \ldots , \sigma _ n : K \to L$ be pairwise distinct morphisms of $F$-extensions. Then $\sigma _1, \ldots , \sigma _ n$ are $L$-linearly independent: if $\lambda _1, \ldots , \lambda _ n \in L$ not all zero, then $\sum \lambda _ i\sigma _ i(\alpha ) \not= 0$ for some $\alpha \in K$.

Proof. Apply Lemma 9.13.1 to the restrictions of $\sigma _ i$ to the groups of units. $\square$

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