Describe the units of $\mathbb {Z}_4[i]$

Question

Describe the units of $\mathbb{Z}_4[i] $?

$\mathbb {Z_4}[i] = \{a+bi, a,b \in \mathbb {Z}_4\}$

$u=a+bi $

$u$ is unit if $N (u) =1$

$a^2+b^2$=1$\implies $

$a=1$, $b=0$ or $b=1$,$a=0 $

so the unit is $u=1$ and $u=i $

Is it true ?

Answer 1

Your base ring is $\Bbb Z_4=\{0,1,2,3\}$, with addition and multiplication calculated modulo $4$. Your extension has an $i$, square root of $-1$. The ring has only sixteen elements, so I think you would find it very useful to write out the full multiplication table.

As to your question, you see that $(i)(3i)=3(-1)=3^2=1$. So the inverse of $i$ is $3i$.

It happens that among those sixteen elements, there are eight units and eight nonunits. Now you go find them.