Any element can be written as where are the equivalence classes mod 3. There are 9 such possibilities just check each one.

A unit in this ring is such that . Thus the pair is invertible if and only if are invertible. Use this to complete the problem.2. Find all units and zero divisors of Z_3 "direct sum" Z_6.

For zero divisors note that if but are in a field and they are no zero divisors hence . But are in so are zero divisors.

The multiplicative set of F of non-zero elements forms a group under multiplication. Since it has n-1 elements it means x^(n-1) = 1 (property of groups).3. Let F be a finite field with n elements. Prove that x^(n-1) = 1 for all nonzero x in F.

Just check the defitions.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).