          # 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.

## Basic definitions

The Gaussian integers are the set

$\mathbf {Z} [i]=\{a+bi\mid a,b\in \mathbf {Z} \},\qquad {\text{ where }}i^{2}=-1.$ In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers. Since the Gaussian integers are closed under addition and multiplication, they form a commutative ring, which is a subring of the field of complex numbers. It is thus an integral domain.

When considered within the complex plane, the Gaussian integers constitute the 2-dimensional integer lattice.

The conjugate of a Gaussian integer a + bi is the Gaussian integer abi.

The norm of a Gaussian integer is its product with its conjugate.

$N(a+bi)=(a+bi)(a-bi)=a^{2}+b^{2}.$ The norm of a Gaussian integer is thus the square of its absolute value as a complex number. The norm of a Gaussian integer is a nonnegative integer, which is a sum of two squares. Thus a norm cannot be of the form 4k + 3, with k integer.

The norm is multiplicative, that is, one has

$N(zw)=N(z)N(w),$ for every pair of Gaussian integers z,w. This can be shown either by a direct check, or by using the multiplicative property of the absolute value of complex numbers.

The units of the ring of Gaussian integers (that is the Gaussian integers whose multiplicative inverse is also a Gaussian integer) are precisely the Gaussian integers with norm 1, that is, 1, –1, i and i.

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