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 $\u2102$; 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

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ordinary prime numbers^{} $\equiv 3mod4$

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

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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  20130322 11:45:32 
Last modified on  20130322 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 5500 
Classification  msc 55U05 
Classification  msc 32M10 
Classification  msc 32C11 
Classification  msc 1402 
Classification  msc 1800 
Related topic  EisensteinIntegers 