Prove that if $z \in \mathbb{Q}(\alpha)$ is nonzero then there exists $z^{-1} \in \mathbb{Q}(\alpha)$ s.t. $z*z^{-1} = 1$.
Prove that the set $\mathbb{Q}[\alpha] = \{a+b\alpha+c\alpha^2 \mid a, b, c \in\mathbb{Q}\}$ is closed under addition and multiplication.