Question

To prove that the set of gaussian integers is an integral domain

Answer 1

We have the following Theorem: A non-zero commutative ring is an integral domain if and only if for all aa,bb ≠0≠0 ⟹ab≠0⟹ab≠0.

Now, we need to prove that the Gaussian integers form an integral domain.

Proof: Let Z[i]Z[i] denote the Gaussian Integers, which is a commutative ring. Take z,w∈Z[i]z,w∈Z[i] s.t: z,w≠0z,w≠0 and z=a+ibz=a+ib, w=c+idw=c+id.

Then, zw=(ac−bd)+(ad+bc)i∈Z[i]zw=(ac−bd)+(ad+bc)i∈Z[i]. Since the elements of Z[i]Z[i] are non-zero ⟹zw≠0⟹zw≠0. QED.