# Gelfond’s theorem

Let $\alpha $ and $\beta $ be algebraic over $\mathbb{Q}$, with $\beta $ irrational and $\alpha $ not equal to 0 or 1. Then ${\alpha}^{\beta}$ is transcendental over $\mathbb{Q}$.

This is perhaps the most useful result in determining whether a number is algebraic or transcendental.

The theorem is also known as the Gelfond-Schneider Theorem^{}.

Title | Gelfond’s theorem |
---|---|

Canonical name | GelfondsTheorem |

Date of creation | 2013-03-22 13:24:31 |

Last modified on | 2013-03-22 13:24:31 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 9 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 11J81 |

Synonym | Gelfond-Schneider Theorem |

Related topic | LindemannWeierstrassTheorem |

Related topic | Irrational |