Prove that Zi is a fieldSolutionFirst we have to show that Z

Prove that Z[i]/<1 - i> is a field

Solution

First, we have to show that Z[i] / <1 - i> has only two elements namely {[0], [1]}.

Since, a + bi = (a + b) - b(1 - i), this implies
[a + bi] = [a + b] for any a,b in Z.
Since -i(1 - i)^2 = 2, we see that
[a + bi] = [a + b] = [0] or [1].

Now we have to prove that the nonzero element in Z[i] / <1 - i> is invertible:
[1]^(-1) = [1].

Hence Z[i] / <1 - i> is a field.
Thus, Z[i] / <1 - i> is a field with 2 elements.