Gaussian Integers form Integral Domain

Theorem
The ring of Gaussian integers $\struct {\Z \sqbrk i, +, \times}$ is an integral domain.

Proof
The set of complex numbers $\C$ forms a field, which is by definition a division ring.

We have that $\Z \sqbrk i \subset \C$.

So from Cancellable Element is Cancellable in Subset, all non-zero elements of $\Z \sqbrk i$ are cancellable for complex multiplication.

The identity element for complex multiplication is $1 + 0 i$ which is in $\Z \sqbrk i$.

We also have that Complex Multiplication is Commutative.

From Identity of Cancellable Monoid is Identity of Submonoid, the identity element of $\struct {\Z \sqbrk i^*, \times}$ is the same as for $\struct {\C^*, \times}$.

So we have that:
 * $\struct {\Z \sqbrk i, +, \times}$ is a commutative ring with unity
 * All non-zero elements of $\struct {\Z \sqbrk i, +, \times}$ are cancellable.

Hence the result from definition of integral domain.