%%Sample LaTeX file for Math 75 paper.  I am not an expert in LaTeX so by no means do I guarantee that this sample file be the best way to use LaTeX!  However, you should find that everything below works. 

%%As you can tell, you use "%" to start a comment in a LaTeX file.  Comments are useful for making comments to yourself, or telling others who might have occasion to read your file (collaborators, for instance) what you are doing at any given place.

%The below declares the documentclass, which specifies the general form the document will take (article, book, etc), as well as any additional packages which are loaded.  There are hundreds of packages out there, each designed with a different purpose in mind.  Packages contain commands which might be useful; for most of what you need to write it should be sufficient to load the amsfonts and amssymb packages.  (AMS = American Mathematical Society)  I prefer using the amsart document class, as I feel it looks the best.

\documentclass[11pt]{amsart}  %%The [11pt] is an optional entry in the documentclass command, and specifies 11pt font.

\usepackage{amsfonts}
\usepackage{amssymb}

%The \newcommand option is used to define your own commands.  These are primarily used to save time typing.  For example, if I want to write the blackboard bold Q for rational numbers, it is obviously much faster to type "\Q" than "\mathbb Q".  
%For the last command listed below, the "[1]" indicates the number of parameters the command you define needs as input, and then the "#1" in the defined command indicates that the first of the parameters you enter when you call the command should be inserted there.  For example, "\Zmod{8}" would write Z/8Z, with appropriate blackboard type.
\newcommand{\Z}{\mathbb Z}
\newcommand{\R}{\mathbb R}
\newcommand{\Q}{\mathbb Q}
\newcommand{\C}{\mathbb C}
\newcommand{\Fp}{\mathbb F_{p}}
\newcommand{\Zmod}[1]{\Z / #1 \Z}

%Once you've made the preliminary commands above, you start your document proper with "\begin{document}".  
\begin{document}

%The below makes a title for your document.  There are other commands you can call in the title if you want to include more information.
\title{A sample Math 75 paper}
\author{Andrew Yang}
\maketitle

%The \section command creates a section, which continues until the next "\section" command is encountered.  You can also create subsections or subsubsections; see the later parts of the file.

\section{An introductory section}
Here is some sample text.

%To write math in a line, use "$" to enter in-line math mode.  
Here is an in-line mathematical formula: $y = x + 2$.   

%If you have a more complicated mathematical expression (or an important expression you want to draw attention to) and you want to set it off in its own "displayed" space, use the "\[" and "\]$ delimiters.  You can also use the longer "\begin{displaymath}" and "\end{displaymath}".  If you want to label your equation with a number, you use "\begin{equation}" and "\end{equation}".  It's not easy to control the actual labels, but this is intentional, as LaTeX will automatically sequentially number them.  In general you should not need to control the numbering.  It's a good practice to only numerically label the important equations which you might have occasion to reference at future points in the paper.

Here is a displayed formula:

%\log is a command which writes "log" in math mode without looking funny.  \int draws an integral sign, while \frac{#1}{#2} displays a fraction with #1 as the numerator and #2 as the denominator.  The \, is a spacing command, which adds a bit more space than usual.  I like to use this in integrals to set the differential apart from the rest of the integrand.
\[ \log x = \int \frac{1}{x} \, dx. \]

%You end a document with "\end{document}".

I can use some of the macros I defined earlier: $\R, \C, \Q, \Z$.  

%To write superscripts or subscripts, use the ^ and _ commands.  You can nest these if you want.  
We can display subscripts and superscripts: $x^{2}, a_{n}$.  

%You can make subsections or subsubsections.  You shouldn't have need to use subsubsections, and probably not subsections either.  
\subsection{A subsection}
\subsubsection{A subsubsection}

%To cite a reference you have included in a reference file, use the command "\cite{#ref}", where #ref refers to the name of the reference you gave it in the reference file.  The ~ command means there should be a space between the : and the citation and ensures that this space does not break the colon and the reference across different lines.  
I can cite a reference:~\cite{ST:RationalPoints}. 



\section{A new section}
	This is a new section!
	
	
%At the end of your file, you include the bibliography list you will cite (in this case, samplem75.bib), and also the style of your references.  I'm using the amsalpha style, but many others exist.  

\bibliographystyle{amsalpha}
\bibliography{samplem75}
\end{document}

