#+TITLE: Assignment Twelve
#+AUTHOR: Lizzy Hunt
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* 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 $2 \notin (6)$.