Gaussian integers - Knowlpedia

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

Arithmetic

Its units are $1,-1,i,-i$ . The norm $a+bi\mapsto a^2+b^2$ is integral and multiplicative.

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.