$p$-extensionPExtension2005-02-17 16:46:082005-02-17 16:49:31Definitionfield extension% 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{amsthm}
\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
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}
\newtheorem{exa}{Example}
% Some sets
\newcommand{\Nats}{\mathbb{N}}
\newcommand{\Ints}{\mathbb{Z}}
\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\newcommand{\Rats}{\mathbb{Q}}\begin{defn}
Let $p$ be a prime number. A Galois extension of fields $E/F$, with $G=\operatorname{Gal}(E/F)$, is said to be a $p$-extension if $G$ is a $p$-group.
\end{defn}
\begin{exa}
Let $d$ be a square-free integer. Then the field extension $\Rats(\sqrt{d})/\Rats$ is a $2$-extension.
\end{exa}
\begin{exa}
Let $p>2$ be a prime and, for any $n$, let $\zeta_{p^n}$ be a primitive $p^n$th root of unity. The cyclotomic extension:
$$\Rats(\zeta_{p^n})/\Rats(\zeta_p)$$
is a $p$-extension. Indeed:
$$G_n=\operatorname{Gal}(\Rats(\zeta_{p^n})/\Rats)\cong (\Ints/p^n\Ints)^\times$$
Thus, $|G_n|=\varphi(p^n)=p^{(n-1)}(p-1)$ and $|G_1|=\varphi(p)=p-1$, where $\varphi$ is the Euler phi function. Therefore the extension above is of degree $p^{(n-1)}$.
\end{exa}