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.

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Gaussian integer definition
Gaussian prime definition

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A prime is a Gaussian prime if and only if it is 3 modulo 4

Tag	`0GTEHJQRFK`
URI	`m2://0GTEHJQRFK`
Kind	definition
Writing	W1 stated
MSC	11R11
Defines	Gaussian prime
Uses	Gaussian integer

Gaussian prime

Definition

Remarks

Reading path

Where this leads

Further reading

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