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 $Z [i]$ . This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.

Formally, the Gaussian integers are the set

Note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the $2$ -dimensional integer lattice.

The (arithmetic or field) norm of a Gaussian integer is the square of its absolute value (Euclidean norm) as a complex number. It is the natural number defined as

where $\cdot$ (an overline) is complex conjugation.

The norm is multiplicative, since the absolute value of complex numbers is multiplicative, i.e., one has

The latter can also be verified by a straightforward check. The units of $Z [i]$ are precisely those elements with norm $1$ , i.e. the set ${\pm1, \pm i}.$

The Gaussian integers form a principal ideal domain with units ${\pm1, \pm i}.$ For $x \in Z [i]$ , the four numbers $\pm x, \pm ix$ are called the associates of $x$ . As for every principal ideal domain, $Z [i]$ is also a unique factorization domain. It follows that a Gaussian integer is prime if and only if it is irreducible.

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