# Algebraic closure not even finitely generated over base field [closed]

Show the algebraic closure $$\bar{\mathbb{Q}}$$ is not even finitely generated over the field $$\mathbb{Q}$$

## closed as off-topic by Servaes, Vinyl_cape_jawa, max_zorn, Leucippus, Parcly TaxelMar 1 at 6:33

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Hint: Show that for any natural $$n$$, there is an algebraic extension of $$\Bbb Q$$ of degree (at least) $$n$$. And any finitely generated algebraic extension has some finite degree.
• @Christian No contradiction at all. Just proving $[\overline{\Bbb Q}:\Bbb Q]\geq n$ for all $n$. That's all we need. – Arthur Mar 1 at 0:56