% Created 2023-04-23 Sun 13:45
% Intended LaTeX compiler: pdflatex
\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{longtable}
\usepackage{wrapfig}
\usepackage{rotating}
\usepackage[normalem]{ulem}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{capt-of}
\usepackage{hyperref}
\notindent \notag \usepackage{ dsfont } \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} \usepackage{polynom} \usepackage{wasysym}
\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)}, 
 pdflang={English}}
\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}