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 $\mathbb{C}$; 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 $\equiv 3\mod 4$

• elements of the form $a\pm bi$ where $a^{2}+b^{2}$ is an ordinary prime $\equiv 1\mod 4$ (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