Construct a multiplication table for $\mathbb{Z}_2[i]$. Construct a multiplication table for $$\mathbb{Z}_2[i]=\{a+bi|a,b \in \mathbb{Z}_2\}$$.
I have figured this table out for the most part. I know that each side has $0,1,i,1+i$, but $(1+i)(i)=i-1$, and what would that become in $\mathbb{Z}_2[i]$? I have the same question for $(1+i)(1+i)$, I don't know how to reduce these in terms of the mod.
As others also mentioned , we can generate the field ${Z_2[x]}$ with irreducible polynomial ${x^2+x+1}$ ,and the elements are ${\{0,1,x,x+1\}}$  , here ${x^2=x+1}$,and  ${-1=1}$ , ${0=(1+i)^2 = i+i}$

Here ${(x+1)^{-1}=x}$, cayley table may looks like..
As others also mentioned , we can generate the field ${Z_2[x]}$ with irreducible polynomial ${x^2+x+1}$ ,and the elements are ${\{0,1,x,x+1\}}$  , here ${x^2=x+1}$,and  ${-1=1}$ , ${0=(1+i)^2 = i+i}$

Here ${(x+1)^{-1}=x}$, cayley table may looks like..