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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+#+TITLE: Assignment Twelve
+#+AUTHOR: Lizzy Hunt
+#+STARTUP: entitiespretty fold inlineimages
+#+LATEX_HEADER: \notindent \notag \usepackage{ dsfont } \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} \usepackage{polynom} \usepackage{wasysym}
+#+LATEX: \setlength\parindent{0pt}
+#+OPTIONS: toc:nil
+
+* Section 6.3
+** Question One
+$n = cd$ 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.
+
+** Question Five
+Both $\mathds{Z}_6$ and $\mathds{Z}_{12}$'s maximal ideals are $(2)$ and $(3)
+** Question Six
+*** a
+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 $(3)$.
+
+*** b
+In $\mathds{Z}_{10}$ the maximal ideals are $(2)$ and $(5)$, similarly for $\mathds{Z}_{15}$: $(3)$ and $(5)$.
+
+** Question Eight
+Consider $(2) \cap (3)$ which generates $(6)$, and is not prime in $\mathds{Z}$; $3 \cdot 2 \in (6)$ but $3 \notin (6)$ and $2 \notin (6)$.
+
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