the field extension $ℝ/\mathbb{Q}$ is not finite

Theorem.

Let $L\mathrm{/}K$ be a finite field extension. Then $L\mathrm{/}K$ is an algebraicextension.

Corollary.

The extension of fields $\mathrm{R}\mathrm{/}\mathrm{Q}$ is not finite.

Proof of the Corollary.

If the extension was finite, it would be an algebraic extension. However, theextension $ℝ/\mathbb{Q}$ is clearly not algebraic. For example,$e\in ℝ$ is transcendental over $\mathbb{Q}$ (see e is transcendental).∎