Lemma 9.13.2. Let $L$ be a field. Let $n \geq 1$ and $\alpha _1, \ldots , \alpha _ n \in L$ pairwise distinct elements of $L$. Then there exists an $e \geq 0$ such that $\sum _{i = 1, \ldots , n} \alpha _ i^ e \not= 0$.

**Proof.**
Apply linear independence of characters (Lemma 9.13.1) to the monoid homomorphisms $\mathbf{Z}_{\geq 0} \to L$, $e \mapsto \alpha _ i^ e$.
$\square$

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