Units of Gaussian Integers form Group/Proof 1

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

By definition of the imaginary unit $i$:

\(\ds i^2\)

\(=\)

\(\ds -1\)

\(\ds i^3\)

\(=\)

\(\ds -i\)

\(\ds i^4\)

\(=\)

\(\ds 1\)

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

Thus $\struct {U_\C, \times}$ is by definition a cyclic group of order $4$.

$\blacksquare$