\documentclass[11pt,twoside]{amsart}
\usepackage{amssymb, amsmath, enumerate, palatino, hyperref, xypic, mathabx, epsfig}
\usepackage[margin=1.5in]{geometry}

%%% Theorem environments %%%
\theoremstyle{theorem}
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\theoremstyle{remark}
\newtheorem{rmk}[thm]{Remark}
%%%%%%%%%%%%%%%%%%

%%% Macros %%%%%%%%%%%
\newcommand{\CC}{\mathbb{C}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}

\newcommand{\normal}{\trianglelefteq}
\newcommand{\ul}{\underline}
\newcommand{\action}{\mathrel{\reflectbox{$\righttoleftarrow$}}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Inn}{\operatorname{Inn}}
\newcommand{\Out}{\operatorname{Out}}
\newcommand{\GSet}{G\text{-}\operatorname{Set}}
\newcommand{\mm}[4]{\begin{pmatrix}#1 &#2\\ #3 &#4\end{pmatrix}}
\newcommand{\Syl}{\operatorname{Syl}}
\newcommand{\semi}{\rtimes}
%%%%%%%%%%%%%%%%%%

%%% Fill in title, author, and date info here
\title{My theory}
\author{\'{E}variste Galois}
\date{May 31, 1832}

\begin{document}
\maketitle

%%% Write the main body of your paper below this line and before \end{document}

This is a paragraph.  It contains sentences.  Good mathematical writing is structured like any other good writing:  into paragraphs containing sentences.

This is a new paragraph.  In the \LaTeX~source code, the paragraph break was created by adding an empty line to the code, \emph{i.e.}, by hitting the return key twice.  Oh, and did you notice that it's possible to make \emph{italic} characters, as well as \textbf{bold} and \underline{underlined}?  The primary place you'd want to use italics is in a definition.

\begin{defn}
Let $k\subseteq L$ be a normal, separable field extension, also written $L/k$.  Then the \emph{Galois group} of $L/k$, denoted $\operatorname{Gal}(L/k)$, is the set of field automorphisms of $L$ which fix $k$ pointwise.
\end{defn}

Note how the above definition appears inside of a \emph{theorem environment}.  The theorem environments in this document are initialized in the preamble of the document.  Go find them.  One of the nicest features of \LaTeX~is that it automatically numbers your theorem environments.

\begin{thm}\label{thm:Galois}
If $L/k$ is a finite, normal, separable field extension, then the subgroups of $\operatorname{Gal}(L/k)$ are in bijective correspondence with the subextensions $k\subseteq E\subseteq L$ of $k\subseteq L$.  
\end{thm}

Using \LaTeX's labelling features, we can even automatically reference particular theorems and equations, such as Theorem \ref{thm:Galois}.

Speaking of equations, they can either appear inline ($\ZZ/mn\ZZ \cong \ZZ/m\ZZ\times \ZZ/n\ZZ$ for $(m,n)=1$) or in display mode:
\[
  \ZZ/mn\ZZ \cong \ZZ/m\ZZ\times \ZZ/n\ZZ \text{ for } (m,n)=1.
\]
You can even number and refer to specific equations, such as
\begin{equation}\label{eqn:Z30}
  \ZZ/30\ZZ \cong \ZZ/2\ZZ\times \ZZ/3\ZZ\times \ZZ/5\ZZ.
\end{equation}
Indeed, equation (\ref{eqn:Z30}) is a specific instance of the Chinese Remainder Theorem.

Finally, you'll need a bibliography.  If you have the time, learning BibTeX is well worth the investment.  If you don't, then \LaTeX~has an embedded bibliography system that you can read about in, \emph{e.g.}, \cite{lamport94}.

\begin{thebibliography}{9}

\bibitem{lamport94}
  Leslie Lamport,
  \emph{\LaTeX: a document preparation system}.
  Addison Wesley, Massachusetts,
  2nd edition,
  1994
  
\end{thebibliography}

\end{document}