# Frobenius map

Let $K$ be any field of characteristic^{} $p>0$, and suppose $K$ contains the finite field^{} ${\mathbb{F}}_{q}$ of size $q$, where $q={p}^{r}$. The ${q}^{\mathrm{th}}$ power Frobenius map^{} on $K$ is the map ${\mathrm{Frob}}_{q}:K\u27f6K$ defined by ${\mathrm{Frob}}_{q}(x):={x}^{q}$.

If $K$ is perfect^{}, then ${\mathrm{Frob}}_{q}$ is an automorphism^{} of $K$ which fixes ${\mathbb{F}}_{q}$, and accordingly is a member of the Galois group^{} $\mathrm{Gal}(K/{\mathbb{F}}_{q})$.

Title | Frobenius map |
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Canonical name | FrobeniusMap |

Date of creation | 2013-03-22 12:34:52 |

Last modified on | 2013-03-22 12:34:52 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 6 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 12E20 |

Classification | msc 11T99 |

Related topic | FrobeniusAutomorphism |