Prove that Zi is a fieldSolutionSolution We know already th

Solution

Solution :

We know already that this is a ring.

First, you need to show that Z[i] / <1 - i> = {[0], [1]}.

This, is straightforward, since a + bi = (a + b) - b(1 - i)

==> [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].

Next, the nonzero element in Z[i] / <1 - i> is invertible:

[1]^(-1) = [1].

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

I hope this helps!