I am having some trouble with these problems:

1. Find all solutions of (x^2) - x + 2 = 0 over Z_3[i], where Z_3[i] =

{a + bi, where a, b are an element of Z_3} = {0, 1, 2, i, 1 + i, 2 + i, 2i, 1 + 2i, 2 + 2i}, where i^2 = -1 (the ring of Gaussian integers modulo 3).

2. Find all units and zero divisors of Z_3 "direct sum" Z_6.

3. Let F be a finite field with n elements. Prove that x^(n-1) = 1 for all nonzero x in F.

4. Is Z_2[i] a field? Is it an integral domain? Note that Z_2[i] = {a + bi, where a, b are an element of Z_2} = {0, 1, i, 1 + i}, where i^2 = -1 (the ring of gaussian integers modulo 2).

For #4, I'm fairly certain it's not an integral domain since (1 + i)(1 + i) = 2i = 0 in mod 2, making (1+i) a zero divisor. I'm not sure how to show if it is a field or not though.

Thanks guys.