Definition:Gaussian Prime

Definition

Definition 1

Let $x \in \Z \sqbrk i$ be a Gaussian integer.

$x$ is a Gaussian prime if and only if:

it cannot be expressed as the product of two Gaussian integers, neither of which is a unit of $\Z \sqbrk i$ (that is, $\pm 1$ or $\pm i$)

it is not itself a unit of $\Z \sqbrk i$.

Definition 2

A Gaussian prime is a Gaussian integer which has exactly $8$ divisors which are themselves Gaussian integers.

Examples

The following are examples of Gaussian primes:

$1 + i$

$4 - i$

$7 + 2 i$

Also see

• Equivalence of Definitions of Gaussian Prime

• Definition:Composite Gaussian Integer

• Results about Gaussian primes can be found here.

Source of Name

This entry was named for Carl Friedrich Gauss.

Historical Note

The concept of a Gaussian prime was introduced by Carl Friedrich Gauss in his papers of $1828$ and $1832$.