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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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