# Math Help - The elements of a field

1. ## The elements of a field

Hi

How to list the elements of the field Z2[x]/<x^2+x+1>?
and how many elements are in Z5[i]/<1+i> ?
thx for the help

2. Originally Posted by toxiccan
Hi

How to list the elements of the field $\frac{\mathbb{Z}_2[x]}{}$?
let $=I.$ then $\frac{\mathbb{Z}_2[x]}{I}=\{a+bx + I: \ a,b \in \mathbb{Z}_2 \} = \{I,1+I,x+I,1+x+I\} \simeq \mathbb{F}_4.$

and how many elements are in $\frac{\mathbb{Z}_5[i]}{<1+i>}$ ?
since in $\mathbb{Z}_5[i]: \ 1=(3+2i)(1+i),$ we have $<1+i>=\mathbb{Z}_5[i]$ and thus the ring has only one element, i.e. $\bar{0}.$