Question regarding algebraicity of two elements whose sum and product are algebraic.

Question

Let $\alpha , \beta \in \Bbb C$ and suppose $\alpha + \beta$ and $\alpha \beta$ are algebraic over $\Bbb Q$. Prove $\alpha , \beta$ are algebraic over $\Bbb Q$.

Answer 1

Observe that $\alpha^2-(\alpha+\beta)\alpha + \alpha \beta = 0$. Hence, $\alpha$ is algebraic over $\mathbb{Q}(\alpha+\beta,\alpha\beta)$. But $\mathbb{Q}(\alpha+\beta,\alpha\beta)$ is algebraic over $\mathbb{Q}$. Hence, $\alpha$ is algebraic over $\mathbb{Q}$.