From 6bf4b90c90f15f4ab60833bddf5b5756d1a6b1f6 Mon Sep 17 00:00:00 2001
From: Elizabeth Alexander Hunt <me@liz.coffee>
Date: Thu, 2 Jul 2026 11:55:17 -0700
Subject: Init

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+% 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}
\ No newline at end of file
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