Why is $[\overline{\mathbb{Q}}:\mathbb{Q}]$ infinite?

Question

I hoped the question had been already asked, but I didn't find it: Why is the index of $\mathbb{Q}$ in its algebraic closure $\overline{\mathbb{Q}}$ infinite?

I am aware that it can be viewed as a consquence of Artin-Schreier theorem, but is there a more elementary way?

Answer 1

For every $n$, $x^n-2$ is irreducible over $\Bbb{Q}$. So if the degree is finite, it has to be divisible by every positive integer.