Units of Gaussian Integers form Group/Proof 2

Theorem

Let $U_\C$ be the set of units of the Gaussian integers:

$U_\C = \set {1, i, -1, -i}$

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

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

Then $\struct {U_\C, \times}$ forms a cyclic group under complex multiplication.

Proof

From Gaussian Integer Units are 4th Roots of Unity:

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

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

$\blacksquare$