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 $1$.

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.

$\blacksquare$

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