Gaussian integer

A complex number of the form $a+bi$, where $a,b\in \mathbb{Z}$, is called a Gaussian integer.

It is easy to see that the set $S$ of all Gaussian integers is a subring of $ℂ$; specifically, $S$ is the smallest subring containing $\{1,i\}$, whence $S=\mathbb{Z}[i]$.

$\mathbb{Z}[i]$ is a Euclidean ring, hence a principal ring, hence a unique factorization domain.

There are four units (i.e. invertible elements) in the ring $\mathbb{Z}[i]$, namely $\pm 1$ and $\pm i$. Up to multiplication by units, the primes in $\mathbb{Z}[i]$ are

• ordinary prime numbers $\equiv 3mod4$

• elements of the form $a\pm bi$ where $a^2+b^2$ is an ordinary prime $\equiv 1mod4$ (see Thue’s lemma)

• the element $1+i$.

Using the ring of Gaussian integers, it is not hard to show, for example, that the Diophantine equation $x^2+1=y^3$ has no solutions $(x,y)\in \mathbb{Z}\times \mathbb{Z}$ except $(0,1)$.

Title	Gaussian integer
Canonical name	GaussianInteger
Date of creation	2013-03-22 11:45:32
Last modified on	2013-03-22 11:45:32
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	11
Author	Daume (40)
Entry type	Definition
Classification	msc 11R04
Classification	msc 55-00
Classification	msc 55U05
Classification	msc 32M10
Classification	msc 32C11
Classification	msc 14-02
Classification	msc 18-00
Related topic	EisensteinIntegers