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

###### Theorem.

Let $L/K$ be a finite field extension. Then $L/K$ is an algebraic extension.

###### Corollary.

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

###### Proof of the Corollary.

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

Title the field extension $\mathbb{R}/\mathbb{Q}$ is not finite TheFieldExtensionmathbbRmathbbQIsNotFinite 2013-03-22 13:57:32 2013-03-22 13:57:32 alozano (2414) alozano (2414) 6 alozano (2414) Corollary msc 12F05 the reals is not a finite extension of the rationals Pi Algebraic FiniteExtension