Gaussian Integer Units form Multiplicative Subgroup of Complex Numbers

Theorem

The group of Gaussian integer units under complex multiplication:

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

forms a subgroup of the multiplicative group of complex numbers.

Proof

By Units of Gaussian Integers form Group, $\struct {U_\C, \times}$ forms a group.

Each of the elements of $U_\C$ is a complex number, and non-zero, and therefore $U_\C \subseteq \C \setminus \set 0$.

The result follows by definition of subgroup.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 36$: Subgroups: Simple illustrations: $(2)$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): subgroup