# example of nonperfect field

In this entry, we exhibit an example of a field that is not a perfect field^{}.

Let $F={\mathbb{F}}_{p}(t)$, where ${\mathbb{F}}_{p}$ is the field with $p$ elements and $t$ transcendental^{} over ${\mathbb{F}}_{p}$. The splitting field^{} $E$ of the irreducible polynomial^{} $f={x}^{p}-t$ is not separable^{} over $F$. Indeed, if $\alpha $ is an element of $E$ such that ${\alpha}^{p}=t$, we have

$${x}^{p}-t={x}^{p}-{\alpha}^{p}={(x-\alpha )}^{p},$$ |

which shows that $f$ has one root of multiplicity $p$.

Title | example of nonperfect field |
---|---|

Canonical name | ExampleOfNonperfectField |

Date of creation | 2013-03-22 13:08:31 |

Last modified on | 2013-03-22 13:08:31 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 7 |

Author | CWoo (3771) |

Entry type | Example |

Classification | msc 12F10 |