\documentclass[a4paper,10pt]{amsart}
\usepackage{amsmath,amssymb,amsthm}

% Fonts
\newcommand{\mf}{\mathfrak}
\newcommand{\mc}{\mathcal}

% Main Rings
\newcommand{\R}{\mathbf R}
\newcommand{\C}{\mathbf C}
\newcommand{\Q}{\mathbf Q}
\newcommand{\Z}{\mathbf Z}
\newcommand{\F}{\mathbf F}

% Number Fields
\newcommand{\ok}{\mc{O}_K}

% Numbering results
\numberwithin{equation}{section}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{theorem*}{Theorem}
\newtheorem{corollary*}{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{remark}[theorem]{Remark}

\begin{document}

\title{Your project topic}
\author{Your name}
\date{\today}

\begin{abstract}
Write a brief summary of your paper here.
\end{abstract}

\maketitle
\section*{Introduction}
Make an introduction. Tell what you will do in the sections of your paper. 

\section{First section}
You can give an equation like this
\begin{equation}\label{w}
 \alpha = \left\lbrace \begin{array}{cl}
       \frac{\sqrt{d_K}}{2} &  \text{ if } d_K \equiv 0 \pmod{4} \\
       \frac{\sqrt{d_K}+1}{2} & \text{ otherwise}
     \end{array}\right.
\end{equation}
This element $\alpha$ is an algebraic integer since it is a root of a monic
polynomial with integer coefficients.

\begin{theorem}\label{quad}
 Let $K$ be a quadratic extension and let $w=(\sqrt{d_K}+d_K)/2$. Then
\[ \ok = \Z[w]. \]
\end{theorem}
\begin{proof}
 A proof this can be found in \cite[Chap.~2]{marcus}. Or you can refer to the
previous equations such as the equation (\ref{w}). Do this by labels not with
numbers!
\end{proof}
Observe that we can define \LaTeX commands in the beginning of our tex file not
to repeat common expressions. For example we can use \textbackslash ok, instead
of the longer expression \textbackslash mathcal\{O\}\underline{\ }K. Please
analyse the file sample.tex.

\section{Second section}
You can refer to you previous results by labels as well. For example, a
corollary of Theorem~\ref{quad} is the following.
\begin{corollary}
 Any quadratic extension has a power basis.
\end{corollary}

\section{Third section}
This an another section. After this section you see the bibliography. Each item
has a label too. Please add your own references. You may use
\texttt{mathscinet}.

\bibliographystyle{amsplain}
\begin{thebibliography}{99}

\bibitem{rosen}
K.~Ireland, M.~Rosen, \textit{A Classical Introduction to Modern Number Theory}.
Second edition. Graduate Texts in Mathematics, 84. Springer-Verlag, New York,
1990.

\bibitem{lang}
S.~Lang, \textit{Elliptic Functions}. Second edition. Graduate Texts in
Mathematics, 112. Springer-Verlag, New York, 1987.

\bibitem{marcus}
D.~A.~Marcus, \textit{Number fields}. Universitext. Springer-Verlag, New
York-Heidelberg, 1977.

\bibitem{washington} L.~C.~Washington,  \textit{Introduction to cyclotomic
fields}. Second edition. Graduate Texts in Mathematics, 83. Springer-Verlag, New
York, 1997.

\end{thebibliography}

\end{document}
