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'Frobenius map'
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| Title of object: |
Frobenius map |
| Canonical Name: |
FrobeniusMap |
| Type: |
Definition |
| Created on: |
2002-04-14 19:42:11-04 |
| Modified on: |
2002-05-03 18:52:04-04 |
| Classification: |
msc:12E20, msc:11T99 |
Preamble:
% 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
\newcommand{\Frob}{\operatorname{Frob}} |
Content:
| 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 $q^{\rm th}$ power Frobenius map on $K$ is the map $\Frob_q: K \longrightarrow K$ defined by $\Frob_q(x) := x^q$. It is an automorphism of $K$ which fixes $\mathbb{F}_q$, and accordingly is a member of the Galois group $\operatorname{Gal}(K/\mathbb{F}_q)$. |
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