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

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.

• Also I have absolutely no idea how to title this question. Help would be appreciated. Nov 5 '20 at 22:40
• How about: "Every element of $\Bbb Q[\alpha]$ is algebraic over $\Bbb Q$ for $\alpha$ algebraic" ?
– user239203
Nov 5 '20 at 22:41

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