Eisenstein integers

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

where a and b are integers and

is a primitive (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 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 ${\displaystyle a+b\omega }$ and ${\displaystyle c+d\omega }$ is given explicitly by

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