# Frobenius homomorphism

Let $F$ be a field of characteristic^{} $p>0$. Then for any $a,b\in F$,

${(a+b)}^{p}$ | $=$ | ${a}^{p}+{b}^{p},$ | ||

${(ab)}^{p}$ | $=$ | ${a}^{p}{b}^{p}.$ |

Thus the map

$$\begin{array}{ccc}\hfill \varphi :F\hfill & \hfill \to \hfill & \hfill F\hfill \\ \hfill a\hfill & \hfill \mapsto \hfill & \hfill {a}^{p}\hfill \end{array}$$ |

is a field homomorphism, called the *Frobenius homomorphism*, or simply the *Frobenius map* on $F$.
If it is surjective then it is an automorphism^{}, and is called the *Frobenius automorphism ^{}*.

Note: This morphism^{} is sometimes also called the “small Frobenius” to distinguish it from the map $a\mapsto {a}^{q}$, with $q={p}^{n}$. This map is then also referred to as the “big Frobenius” or the “power Frobenius map”.

Title | Frobenius homomorphism |
---|---|

Canonical name | FrobeniusHomomorphism |

Date of creation | 2013-03-22 12:22:50 |

Last modified on | 2013-03-22 12:22:50 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 11 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 12E99 |

Synonym | Frobenius endomorphism |

Synonym | Frobenius map |

Related topic | FrobeniusMorphism |

Related topic | FrobeniusMap |

Defines | Frobenius automorphism |