Let $a \in \mathbf{Q}^{\times}$ be a nonzero rational number. Deduce from Theorem 1 that, for every nonzero polynomial $P \in \mathbf{Q}[x]$, we have $P(e^a) \neq 0$.
(Theorem 1: Let $r \geq 2$ be an integer. If $a_1, \ldots, a_r \in \mathbf{Q}$ are distinct rational numbers, then the real numbers $e^{a_1}, \ldots, e^{a_r}$ are linearly independent over $\mathbf{Q}$.)