Units of Gaussian Integers form Group

Theorem
The set of complex numbers:
 * $U_\C = \left\{{1, i, -1, -i}\right\}$

forms a cyclic group under complex multiplication.

Proof
Elements of $U_\C$ are the units of the Gaussian integers.

From Group of Units is Group, $\left({U_\C, \times}\right)$ forms a group.

It remains to note that:

thus demonstrating that $U_\C$ is generated by $i$.

Hence the result, by definition of cyclic group.