# simple transcendental field extension

The extension field^{} $K(\alpha )$ of a base field^{} $K$, where $\alpha $ is a transcendental element with respect to $K$, is a *simple (http://planetmath.org/SimpleFieldExtension) transcendental extension ^{} of* $K$. All such extension fields are isomorphic to the field $K(X)$ of rational functions in one indeterminate

^{}$X$ over $K$, and thus to each other.

Example. The subfields^{} $\mathbb{Q}(\pi )$ and $\mathbb{Q}(e)$ of $\mathbb{R}$, where $\pi $ is Ludolph’s constant (http://planetmath.org/Pi) and $e$ Napier’s constant, are isomorphic.

Title | simple transcendental field extension |
---|---|

Canonical name | SimpleTranscendentalFieldExtension |

Date of creation | 2013-03-22 15:02:20 |

Last modified on | 2013-03-22 15:02:20 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 11 |

Author | pahio (2872) |

Entry type | Corollary |

Classification | msc 12F99 |

Synonym | simple infinite field extension |

Related topic | FunctionField |