Gaussian prime

Definition

A Gaussian prime is a prime element of the ring $\mathbb{Z}[i]$ of Gaussian integers. That is, it is a nonzero Gaussian integer $\pi$ that is not a unit, with the following property. For all Gaussian integers $\alpha$ and $\beta$, if $\pi \mid \alpha \beta$, then $\pi \mid \alpha$ or $\pi \mid \beta$.

Remarks

Since $\mathbb{Z}[i]$ is a unique factorization domain, the Gaussian primes are also the irreducible elements: the non-units that are not a product of two non-units. A Gaussian integer $\alpha$ is a Gaussian prime if and only if one of two conditions holds. Either $N(\alpha)$ is a prime number, or $\alpha$ is a unit times a prime $p \equiv 3 \pmod 4$. For example, $1 + i$, $2 + i$ and $3$ are Gaussian primes. But $5 = (2 + i)(2 - i)$ is not.

Explanations

L1 human

Gaussian primes play the role of prime numbers in the Gaussian integers. Some ordinary primes, such as $5$, split into two Gaussian primes. Others, such as $3$, stay prime.