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%Header LaTeX page
%Has the following short-hands:
% intersect, union, results, if and only if, modulo, angle
% all usual sets
%
% page settings

\usepackage{latexsym, amsfonts, graphicx, amsthm}



%\setlength{\textheight}{8.75in} \setlength{\textwidth}{6.5in}
%\setlength{\topmargin}{0.0in} \setlength{\headheight}{0.0in}
%\setlength{\headsep}{0.0in} \setlength{\leftmargin}{0.0in}
%\setlength{\oddsidemargin}{0.0in}
\hoffset=-0.05\textwidth
\textwidth=1.1\textwidth
\voffset=-0.05\textheight
\textheight=1.1\textheight

%Enumeration
  \def\be{\begin{enumerate}}
  \def\ee{\end{enumerate}}
  \def\ii{\item}
  \def\bi{\begin{itemize}}
  \def\ei{\end{itemize}}

%Equations and Formatting
  \def\beqa{\begin{eqnarray*}}
  \def\eeqa{\end{eqnarray*}}
  \def\dsp{\displaystyle}

\newcommand{\rez}{\Longrightarrow}
\newcommand{\N}{\mathbf{N}}
\newcommand{\Z}{\mathbf{Z}}
\newcommand{\Q}{\mathbf{Q}}
\newcommand{\R}{\mathbf{R}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\CC}{\mathbf{C}}
\newcommand{\m}{{\textmd{mod}\,\,}}
\renewcommand{\a}{\angle}
\renewcommand{\d}{\Delta}
\renewcommand{\to}{\longrightarrow}
\renewcommand{\v}{\emptyset}
\renewcommand{\u}{$\cup$}
\newcommand{\intp}[1]{\lfloor {#1} \rfloor}
\newcommand{\binom}[2]{{{#1} \choose {#2}}}
\newcommand{\dg}{^{\circ}}

\newcommand{\vv}[1]{{#1}_1,{#1}_2,\ldots ,{#1}_n}
\newcommand{\vvi}[1]{{#1}_1,{#1}_2,\ldots}
\newcommand{\vvn}[2]{{#1}_1,{#1}_2,\ldots ,{#1}_{#2}}

\newcommand{\bea}{\begin{eqnarray*}}
\newcommand{\eea}{\end{eqnarray*}}
\newcommand{\bean}{\begin{eqnarray}}
\newcommand{\eean}{\end{eqnarray}}

\newcommand{\Gal}{\textrm{Gal}}
\newcommand{\GL}{\textrm{GL}}
\newcommand{\SL}{\textrm{SL}}

\def\divides{\ |\ }
\def\dvd{\divides}
\def\cis{\hbox{cis }}
\def\ndvd{\,\not|\ }
\def\natchp{{\bf N'}}
\def\proof{\par{\bf Proof: }}
\def\soln{\par{\bf Solution: }}
\def\hint{{\bf  Hint: }}
\def\endpf{\hfill \vrule depth-1pt height7pt width6pt}
\font\ttl=cmr10 scaled \magstep 3
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{thmfr}[theorem]{Th\'{e}or\`{e}me}


\renewcommand{\iff}{\Longleftrightarrow}