# algebraically closed

A field $K$ is *algebraically closed ^{}* if every non-constant polynomial

^{}in $K[X]$ has a root in $K$.

An extension field^{} $L$ of $K$ is an *algebraic closure* of $K$ if $L$ is algebraically closed and every element of $L$ is algebraic over $K$. Using the axiom of choice^{}, one can show that any field has an algebraic closure. Moreover, any two algebraic closures of a field are isomorphic as fields, but not necessarily canonically isomorphic.

Title | algebraically closed |
---|---|

Canonical name | AlgebraicallyClosed |

Date of creation | 2013-03-22 12:12:06 |

Last modified on | 2013-03-22 12:12:06 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 10 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 12F05 |

Defines | algebraic closure |