In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form

where a and b are integers and

is a primitive (hence non-real) cube root of unity.

The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. The Eisenstein integers are a countably infinite set.

The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field Q(ω) – the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each z = a + bω is a root of the monic polynomial

In particular, ω satisfies the equation

The product of two Eisenstein integers a + bω and c + dω is given explicitly by

The 2-norm of an Eisenstein integer is just its squared modulus, and is given by