Frobenius homomorphism FrobeniusAutomorphism 2002-02-18 23:10:22 2006-10-04 09:00:05 Definition Frobenius automorphism % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $F$ be a field of characteristic $p>0$. Then for any $a, b \in F$, \begin{eqnarray*} (a + b)^p &=& a^p + b^p, \\ (ab)^p &=& a^p b^p. \end{eqnarray*} Thus the map $$ \begin{matrix}\phi: F &\to& F \\ a &\mapsto& a^p \end{matrix} $$ is a field homomorphism, called the \emph{Frobenius homomorphism}, or simply the \emph{Frobenius map} on $F$. If it is surjective then it is an automorphism, and is called the \emph{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''.