Units of Gaussian Integers form Group/Proof 2

Theorem
Let $U_\C$ be the set of units of the Gaussian integers:
 * $U_\C = \left\{{1, i, -1, -i}\right\}$

where $i$ is the imaginary unit: $i = \sqrt {-1}$.

Let $\left({U_\C, \times}\right)$ be the algebraic structure formed by $U_\C$ under the operation of complex multiplication.

Then $\left({U_\C, \times}\right)$ forms a cyclic group under complex multiplication.

Proof
We have that:

Thus $\left\{{1, i, -1, -i}\right\}$ constitutes the set of the $4$th roots of unity.

The result follows from Roots of Unity under Multiplication form Cyclic Group.