# minimal polynomial

Let $K/F$ be a field extension and $\kappa \in K$ be algebraic over $F$. The *minimal polynomial for $\kappa $ over $F$* is a monic polynomial $m(x)\in F[x]$ such that $m(\kappa )=0$ and, for any other polynomial^{} $f(x)\in F[x]$ with $f(\kappa )=0$, $m$ divides $f$. Note that, for any element $\kappa $ that is algebraic over $F$, a minimal polynomial exists (http://planetmath.org/ExistenceOfTheMinimalPolynomial); moreover, because of the monic condition, it exists uniquely.

Given $\kappa \in K$, a polynomial $m$ is the minimal polynomial of $\kappa $ if and only if $m(\kappa )=0$ and $m$ is both monic and irreducible (http://planetmath.org/IrreduciblePolynomial).

Title | minimal polynomial |
---|---|

Canonical name | MinimalPolynomial |

Date of creation | 2013-03-22 13:20:11 |

Last modified on | 2013-03-22 13:20:11 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 13 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 11C08 |

Classification | msc 11R04 |

Classification | msc 12F05 |

Classification | msc 12E05 |

Related topic | DegreeOfAnAlgebraicNumber |