algebraic_construction.tex

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Start document
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\begin{titlepage}
  \begin{center}
    ~\\
    \vspace{6em}
    {\fontsize{34}{48}\selectfont\textsc{Algebraic Construction}}
    ~\\
    \vspace{2.5em}
    {\Large }
    ~\\
    \vspace{6em}
    \textsf{\Large Huyi Chen}
    ~\\
    \vspace{5in}
    {\large Latest Update: \today}
  \end{center}
\end{titlepage}

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% table of contents

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\tableofcontents
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% Your document content here

\chapter*{Notation Conventions}

In this book, we use the following notation conventions:
\begin{itemize}
  \item $\mathbb{N}$: the set of natural numbers $\{0,1,2,\cdots\}$.
\end{itemize}
We use sans-serif font for categories. Some common categories are
\begin{itemize}
  \item $\mathsf{FinSet}$: the category of finite sets.
  \item $\mathsf{Set}$: the category of sets.
  \item $\mathsf{Mon}$: the category of monoids.
  \item $\mathsf{Grp}$: the category of groups.
  \item $\mathsf{Ab}$: the category of abelian groups.
  \item $\mathsf{Ring}$: the category of rings.
  \item $\mathsf{CRing}$: the category of commutative rings.
  \item $\mathsf{Field}$: the category of fields.
  \item $R\text{-}\mathsf{Mod}$: the category of left $R$-modules, where $R\in \mathrm{Ob}\left(\mathsf{Ring}\right)$.
  \item $K\text{-}\mathsf{Vect}$: the category of $K$-vector spaces, where $K\in \mathrm{Ob}\left(\mathsf{Field}\right)$.
  \item $R\text{-}\mathsf{Alg}$: the category of associative $R$-algebras, where $R\in \mathrm{Ob}\left(\mathsf{CRing}\right)$.
  \item $R\text{-}\mathsf{CAlg}$: the category of commutative $R$-algebras, where $R\in \mathrm{Ob}\left(\mathsf{CRing}\right)$.
  \item $\mathsf{Top}$: the category of topological spaces.
\end{itemize}


\include{set_theory}
\include{category_theory}
\include{homological_algebra}
\include{group}
\include{ring}
\include{commutative_ring}
\include{module}
\include{associative_algebra}
\include{vector_space}
\include{field}
\include{galois_theory}
\include{valuation_theory}
%\include{number_theory}


\include{appendices}

\printindex

% \begin{Theorem}{Title}{label}
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%     This is the statement of the corollary.
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% \begin{Example}{Title}{label}
%     This is an example.
% \end{Example}

% \begin{Definition}{Title}{label}
%     This is a definition.
% \end{Definition}
\end{document}