Show that $I ( \alpha )$ is an ideal of $\mathbb { Q } [ X ]$, different from $\{ 0 \}$, where $\alpha$ is a fixed algebraic number and $I ( \alpha ) = \{ P \in \mathbb { Q } [ X ] \mid P ( \alpha ) = 0 \}$.
Show that $I ( \alpha )$ is an ideal of $\mathbb { Q } [ X ]$, different from $\{ 0 \}$, where $\alpha$ is a fixed algebraic number and $I ( \alpha ) = \{ P \in \mathbb { Q } [ X ] \mid P ( \alpha ) = 0 \}$.