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}{Senspacearabic{section}}
renewcommand*{dictumwidth}{0.7textwidth}
renewcommand*{raggeddictum}{raggedright}
renewcommand*{raggeddictumtext}{}

addtokomafont{disposition}{rmfamilymdseries}
addtokomafont{chapterprefix}{large}
setkomafont{dictumtext}{normalfontnormalcoloritshapesetlength{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}{rightskip1initshape}{}

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

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