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

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# 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). ∎

Keywords:

pi, transcendental, reals, rationals

Related:

Pi, Algebraic, FiniteExtension

Synonym:

the reals is not a finite extension of the rationals

Major Section:

Reference

Type of Math Object:

Corollary

## Mathematics Subject Classification

12F05*no label found*

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