# separable

An irreducible polynomial^{} $f\in F[x]$ with coefficients^{} in a field $F$ is separable if $f$ factors into distinct linear factors over a splitting field^{} $K$ of $f$.

A polynomial^{} $g$ with coefficients in $F$ is separable if each irreducible^{} factor of $g$ in $F[x]$ is a separable polynomial.

An algebraic field extension $K/F$ is separable if, for each $a\in K$, the minimal polynomial of $a$ over $F$ is separable. When $F$ has characteristic zero, every algebraic extension of $F$ is separable; examples of inseparable extensions^{} include the quotient field $K(u)[t]/({t}^{p}-u)$ over the field $K(u)$ of rational functions in one variable, where $K$ has characteristic^{} $p>0$.

More generally, an arbitrary field extension $K/F$ is defined to be separable if every finitely generated^{} intermediate field extension $L/F$ has a transcendence basis $S\subset L$ such that $L$ is a separable algebraic extension of $F(S)$.

Title | separable |
---|---|

Canonical name | Separable |

Date of creation | 2013-03-22 12:08:04 |

Last modified on | 2013-03-22 12:08:04 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 13 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 12F10 |

Classification | msc 11R32 |

Related topic | PerfectField |

Defines | separable |

Defines | separable polynomial |

Defines | separable extension |