Is factor ring field?

Question

There is defined

Z[i] = {a+bi|a,b $\in$ Z}

with standard operations of addition and multiplications complex number. Question is, if factor ring Z[i]/(1-i) is field. How could I prove it? Do you have any hint?

Answer 1

Approach 1: We have $2=(1-i)(1+i)$, so in $\Bbb Z[i]/(1-i)$, we have $2=1-i=2i=0$. It doesn't take much work from here to see that the ring has two elements, and what the multiplication in this ring is.

Approach 2: We have $\Bbb Z[i]\cong \Bbb Z[x]/(x^2+1)$, so $\Bbb Z[i]/(1-i)\cong\Bbb Z[x]/(x^2+1, 1-x)$. Now figure out which ring this is by dividing out by first $(1-x)$, and then $(x^2+1)$ instead of the other way around.