Editing Gaussian integer (section)

{{Short description|Complex number whose real and imaginary parts are both integers}}
{{distinguish|text=[[Gaussian integral]]}}
In [[number theory]], a '''Gaussian integer''' is a [[complex number]] whose real and imaginary parts are both [[integer]]s. The Gaussian integers, with ordinary [[addition]] and [[multiplication]] of [[complex number]]s, form an [[integral domain]], usually written as <math>\mathbf{Z}[i]</math> or <math>\Z[i].</math><ref name="Fraleigh 1976 286">{{harvtxt|Fraleigh|1976|p=286}}</ref> 

Gaussian integers share many properties with integers: they form a [[Euclidean domain]], and thus have a [[Euclidean division]] and a [[Euclidean algorithm]]; this implies [[unique factorization]] and many related properties. However, Gaussian integers do not have a [[total order]] that respects arithmetic. 

Gaussian integers are [[algebraic integer]]s and form the simplest ring of [[quadratic integer]]s.

Gaussian integers are  named after the German mathematician [[Carl Friedrich Gauss]].

[[File:Gaussian integer lattice.svg|thumb|217px|Gaussian integers as [[lattice point]]s in the [[complex plane]]]]