Gaussian Integer Units are 4th Roots of Unity

Theorem

The units of the ring of Gaussian integers:

$\set {1, i, -1, -i}$

are the (complex) $4$th roots of $1$.

Proof

We have that $i = \sqrt {-1}$ is the imaginary unit.

Thus:

\(\ds 1^4\)

\(\)

\(\ds \)

\(\ds = 1\)

\(\ds i^4\)

\(=\)

\(\ds \paren {-1}^2\)

\(\ds = 1\)

\(\ds \paren {-1}^4\)

\(=\)

\(\ds 1^2\)

\(\ds = 1\)

\(\ds \paren {-i}^4\)

\(=\)

\(\ds \paren {-1}^2 \cdot \paren {-1}^2\)

\(\ds = 1\)

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

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gaussian integer
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gaussian integer