Please show all steps in the proof Please make answer in a r

Please show all steps in the proof. Please make answer in a readable format. Thanks.

Solution

It is a known fact that 1i is not a unit, we also do know that 0 and 1 symbolize different cosets. As 2 = (1i)(1+i), all the real integers are in the 0 or 1 coset. Now, if we look at a+bi carefully, we can write a+bi(b)(1i) = a+b, and hence we can claim that everything is contained in <1 i> or 1 + <1 i>

Therefore, factor ring turns out to be just {0, 1}, which is ineed a field.