Gaussian integer

A complex numberMathworldPlanetmathPlanetmath of the form a+bi, where a,b, is called a Gaussian integerMathworldPlanetmath.

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

[i] is a Euclidean ringMathworldPlanetmath, hence a principal ringMathworldPlanetmath, hence a unique factorization domainMathworldPlanetmath.

There are four units (i.e. invertible elements) in the ring [i], namely ±1 and ±i. Up to multiplicationPlanetmathPlanetmath by units, the primes in [i] are

  • ordinary prime numbersMathworldPlanetmath 3mod4

  • elements of the form a±bi where a2+b2 is an ordinary prime 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 equationMathworldPlanetmath x2+1=y3 has no solutions (x,y)× 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