Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as ${\displaystyle \mathbf {Z} [i]}$ or ${\displaystyle \mathbb {Z} [i].}$^[1]

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

Gaussian integers are algebraic integers and form the simplest ring of quadratic integers.

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