Units of Gaussian Integers form Group/Proof 3
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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 Units of Gaussian Integers, $U_\C$ is the set of units of the ring of Gaussian integers.
From Group of Units is Group, $\left({U_\C, \times}\right)$ forms a group.
It remains to note that:
\(\ds i^2\) | \(=\) | \(\ds -1\) | ||||||||||||
\(\ds i^3\) | \(=\) | \(\ds -i\) | ||||||||||||
\(\ds i^4\) | \(=\) | \(\ds 1\) |
thus demonstrating that $U_\C$ is cyclic.
$\blacksquare$