Eisenstein integers - Knowlpedia

The eisenstein integers form the subring $\mathbb Z[\omega]=\{a+b\omega:a,b\in\mathbb Z\}$ of its quadratic rational field, with multiplication determined by $\omega^2+\omega+1=0$ .

Arithmetic

Take $\omega=(-1+\sqrt{-3})/2$ . The norm $a+b\omega\mapsto a^2-ab+b^2$ is integral and multiplicative. Its six units are $\pm1,\pm\omega,\pm\omega^2$ .

Category distinction

Its additive group is a free abelian group of rank two. It is a ring and an integral lattice, but it is not a rational vector space: division by every nonzero integer does not stay in the ring. Tensoring with $\mathbb Q$ gives the associated quadratic field.

References

J. S. Milne, Algebraic Number Theory, Chapter 2, Example 2.41, p. 39: integral bases in quadratic fields.