An 'abstract' at the start of every chapter?

Here is an example that uses one of the KOMA-Script classes to achieve a similar effect:

\documentclass[pagesize=auto, version=last, chapterprefix=true]{scrbook}

\usepackage{amsmath}
\usepackage{amsfonts}

\renewcommand*{\thesection}{\S\enspace\arabic{section}}
\renewcommand*{\dictumwidth}{0.7\textwidth}
\renewcommand*{\raggeddictum}{\raggedright}
\renewcommand*{\raggeddictumtext}{}

\addtokomafont{disposition}{\rmfamily\mdseries}
\addtokomafont{chapterprefix}{\large}
\setkomafont{dictumtext}{\normalfont\normalcolor\itshape\setlength{\parindent}{1em}\noindent}


\begin{document}


\setchapterpreamble{%
  \dictum{%
    The notion of prime number is fundamental in number theory.
    The first part of this chapter is devoted to proving that every integer can be written as a product of primes in an essentially unique way.

    After that, we shall prove an analogous theorem in the ring of polynomials over a field.%
  }%
  \vspace{24pt}%
}

\chapter{Unique Factorization}

\section{Unique Factorization in $\mathbb Z$}


\end{document}

I just use

\begin{quote}
{\small ...}
\end{quote} 

You can define a similar abstract environment to your example using

\newenvironment{abstract}{\rightskip1in\itshape}{}

Just use \begin{abstract}...\end{abstract} after your \chapter and then start a \section after that.