Every element of $\mathbb{Q}[\alpha]$ is algebraic over $\mathbb{Q}$ for an $\alpha$ algebraic

Question

So I am stumped by the following problem:

Let $\alpha \in \mathbb{R}$ be an algebraic number of degree $d$. We denote by $\mathbb{Q}[\alpha]$ the subspace of $\mathbb{R}$ consisting of real numbers that can be written as: $$a_0 + a_1\alpha +\dots+ a_{d-1}\alpha^{d-1}$$ for some rational numbers $a_0, ... ,a_{d-1} \in \mathbb{Q}$.

Show that every element $\gamma \in \mathbb{Q[\alpha]}$ is algebraic.

Does anyone have any pointers? I know the definition of an algebraic number.

Answer 1

Hint $\mathbb Q[\gamma]$ is a subspace of $\mathbb Q[\alpha]$. What can you conclude about $$\mbox{dim}_{\mathbb Q} \mathbb Q[\gamma] \,?$$