# the field extension $\mathbb{R}/\mathbb{Q}$ 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 $\mathbb{R}/\mathbb{Q}$ 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 |
---|---|

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 |