# algebraically dependent

Let $L$ be a field extension of a field $K$. Two elements $\alpha ,\beta $ of $L$ are *algebraically dependent* if there exists a non-zero polynomial^{} $f(x,y)\in K[x,y]$ such that $f(\alpha ,\beta )=0$. If no such polynomial exists, $\alpha $ and $\beta $ are said to be *algebraically independent*.

More generally, elements ${\alpha}_{1},\mathrm{\dots},{\alpha}_{n}\in L$ are said to be algebraically dependent if there exists a non-zero polynomial $f({x}_{1},\mathrm{\dots},{x}_{n})\in K[{x}_{1},\mathrm{\dots},{x}_{n}]$ such that $f({\alpha}_{1},{\alpha}_{2},\mathrm{\dots},{\alpha}_{n})=0$. If no such polynomial exists, the collection^{} of $\alpha $’s are said to be algebraically independent.

Title | algebraically dependent |
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Canonical name | AlgebraicallyDependent |

Date of creation | 2013-03-22 13:58:13 |

Last modified on | 2013-03-22 13:58:13 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 12F05 |

Classification | msc 11J85 |

Related topic | DependenceRelation |

Defines | algebraically independent |

Defines | algebraic dependence |

Defines | algebraic independence |