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% Created 2023-04-23 Sun 13:45
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\author{Lizzy Hunt}
\date{\today}
\title{Assignment Twelve}
\hypersetup{
pdfauthor={Lizzy Hunt},
pdftitle={Assignment Twelve},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 28.2 (Org mode 9.6.1)},
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\begin{document}
\maketitle
\setlength\parindent{0pt}
\section{Section 6.3}
\label{sec:org86da123}
\subsection{Question One}
\label{sec:org600275c}
\(n = cd\) with some \(1 < |c| < |n|\) and \(1 < |d| < |n|\) since \(n\) is composite, so \(c\) and \(d\) are not multiples of \(n\). Therefore as \(cd \in (n)\) but \(c \notin (n)\) and \(d \notin (n)\) then \((n)\) is not a prime ideal
by definition.
\subsection{Question Five}
\label{sec:org37fce42}
Both \(\mathds{Z}_6\) and \(\mathds{Z}_{12}\)'s maximal ideals are \((2)\) and \$(3)
\subsection{Question Six}
\label{sec:org18d9056}
\subsubsection{a}
\label{sec:org0890b14}
The only maximal ideal of \(\mathds{Z}_8\) is \((2)\) since it is its prime divisor.
Similarly, the only maximal ideal of \(\mathds{Z}_9\) is \((3)\).
\subsubsection{b}
\label{sec:org77d60ec}
In \(\mathds{Z}_{10}\) the maximal ideals are \((2)\) and \((5)\), similarly for \(\mathds{Z}_{15}\): \((3)\) and \((5)\).
\subsection{Question Eight}
\label{sec:org38c7c8c}
Consider \((2) \cap (3)\) which generates \((6)\), and is not prime in \(\mathds{Z}\); \(3 \cdot 2 \in (6)\) but \(3 \notin (6)\) and \(2 \notin (6)\).
\end{document}
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