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 algebraic extension.

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, the extension $ℝ/\mathbb{Q}$ is clearly not algebraic. For example, $e\in ℝ$ is transcendental over $\mathbb{Q}$ (see e is transcendental). ∎

Title	the field extension $ℝ/\mathbb{Q}$ is not finite
Canonical name	TheFieldExtensionmathbbRmathbbQIsNotFinite
Date of creation	2013-03-22 13:57:32
Last modified on	2013-03-22 13:57:32
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	6
Author	alozano (2414)
Entry type	Corollary
Classification	msc 12F05
Synonym	the reals is not a finite extension of the rationals
Related topic	Pi
Related topic	Algebraic
Related topic	FiniteExtension