Units of Gaussian Integers form Group

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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 1

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 $4$.

$\blacksquare$


Proof 2

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.

$\blacksquare$


Proof 3

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.

$\blacksquare$


Sources

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