Is $\bar{\mathbb{Q}}/\mathbb{Q}$ finitely generated field extension?

Question

Sorry for my bad English.

I know $\bar{\mathbb{Q}}/\mathbb{Q}$ is infinitely extension. Namely, $[\bar{\mathbb{Q}}:\mathbb{Q}]=\infty.$

Now I want to know is $\bar{\mathbb{Q}}/\mathbb{Q}$ finitely extension as field extension? Namely, are there $\alpha_1,\dots,\alpha_n\in \bar{\mathbb{Q}}$ such that $\bar{\mathbb{Q}}=\mathbb{Q}(\alpha_1,\dots,\alpha_n)$?

Please tell me proof or hint or good book, thanks.

Answer 1

Hint: $\mathbb Q(\alpha_1, ... , \alpha_n) = \mathbb Q(\alpha_1, ... , \alpha_{n-1})(\alpha_n)$. If $F$ is a field, and $\alpha$ is algebraic over $F$, what can you say about $[F(\alpha) : F]$?