\documentclass[12pt]{article} % Cahiers wants Times New Roman: \usepackage{times} % Cahiers wants this page layout: \usepackage[a4paper,text={128mm,185mm}, centering]{geometry} \usepackage{changepage} % Slight modification of section titles (optional): \usepackage{titlesec} \titleformat{\section}[hang]% {\bfseries\large}{\thesection.}{1ex}{}% \titleformat{\subsection}[hang]% {\bfseries}{\thesubsection}{1ex}{}% % Make theorem environments (suggested use, modify as you wish): \usepackage{amsthm} \usepackage{fancyhdr} \pagestyle{fancy} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \theoremstyle{definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} % Add other packages that you need, such as: \usepackage[matrix,arrow,curve,cmtip]{xy} \usepackage{amssymb} \usepackage{latexsym} \usepackage{graphicx} % Here the content of the document starts: %lefthead and rightheads: authors and short title \lhead{\sc\bfseries C. Gauss} \rhead{\sc\bfseries short title} %title \title{\vskip 5pt \bf COMPLETELY COMPLETE VERY LONG TITLE} % Nota Bene : please, write your title in capital letters, as in the example here. %authors \author{\itshape\bfseries { Carl GAUSS}} \date{} \begin{document} \maketitle %Nota bene: the next two commands erase pages numbers (the right page numbers will be add by the editors on the pdf) \cfoot{} \thispagestyle{empty} %% \vskip 25pt % Abstract in French and in English, followed by Keywords and MSC: \begin{adjustwidth}{0.5cm}{0.5cm} {\small {\bf R\'esum\'e.} Dans cet article, nous \'etudions des racines primitives, et deux ou trois autres questions arithm\'etiques.\\ {\bf Abstract.} In this article, we study primitive roots, and two or three problems in arithmetics.\\ {\bf Keywords.} Your keywords come here.\\ {\bf Mathematics Subject Classification (2010).} Your MSC numbers come here. } \end{adjustwidth} % Here starts the text:. \section{Introduction} In this paper, we prove that, for any prime $p\in\mathbb{N}$, the group of units in the quotient ring $\mathbb{Z}/(p)$ is cyclic. This is important in many aspects of number theory. \section{Primitive roots} We start with a definition. \begin{definition} Let $n\in\mathbb{N}$ be any non-zero number. We say that a number $a\in\mathbb{Z}$ is a {\em primitive root modulo $n$} when $a+(n)\in\mathbb{Z}/(n)$ is a generator for the group of units in the quotient ring $\mathbb{Z}/(n)$. \end{definition} Our aim now is to show that, for any prime $p\in\mathbb{N}$, there exists a primitive root modulo $p$. We use a lemma, in which $\varphi$ denotes the {\em Euler phi function}. \begin{lemma} Let $G$ be a finite commutative group (written multiplicatively). Then $G$ is cyclic if and only if, for each divisor $d$ of $|G|$, $$|\{g\in G\mid g\mbox{ is of order }d\}|\leq\varphi(d).$$ \end{lemma} \begin{proof} Write $m=|G|$ and, whenever $d$ divides $m$, $G_d\subseteq G$ for the set of elements of order $d$. By Lagrange's Theorem these form a partition of $G$: $$G=\bigcup_{d|m}G_d.$$ Counting on both sides shows that $m=\sum_{d|m}|G_d|$. If $|G_d|\leq\varphi(d)$ for each divisor $d$ of $m$, then $$m=\sum_{d|m}|G_d|\leq\sum_{d|m}\varphi(d)=m$$ by a well-known property of $\varphi$. Hence necessarily $|G_d|=\varphi(d)$ for each divisor $d$ of $m$, thus in particular $|G_m|=\varphi(m)\neq 0$, saying precisely that $G$ is cyclic. The converse is left to the reader. \end{proof} Without proof we mention the following. \begin{proposition} If $G$ is a finite commutative group, and for every divisor $d$ of $|G|$ we have that $$|\{g\in G\mid g^d=1\}|\leq d,$$ then $G$ is cyclic. \end{proposition} We can now state our main result. \begin{theorem} For every prime $p\in\mathbb{N}$, the group $(\mathbb{Z}/(p))^{\times}$ is cyclic. \end{theorem} \begin{proof} The number of roots of the polynomial $X^d-1$, viewed as polynomial with coefficients in the field $\mathbb{Z}/(p)$, is less than or equal to $d$. The previous proposition applies to the group of units of this field. \end{proof} \begin{example} It is easy to verify that $2$ is a primitive root modulo $11$. \end{example} \begin{remark} For more information on the lay-out of this article for the {\em Cahiers}, the official {\em Guide to Authors} can be found at the following address: \begin{center} {\tt http://ehres.pagesperso-orange.fr/Cahiers/Ctgdc.htm}. \end{center} \end{remark} % List of references \begin{thebibliography}{99} \bibitem{gauss}[C. F. Gauss, 1801] Disquisitiones Arithmeticae. \end{thebibliography} % Cahiers wants the author's address at the end of the paper: \vspace{5mm} \noindent Carl Friedrich Gauss \\ Mathematisches Institut \\ University of G\"ottingen \\ Bunsenstrasse 3-5 \\ 37073 G\"ottingen (Germany) \\ my.email@server.univ.xy % Done! \end{document}