The field of Gaussian rationals provides an example of an algebraic number field, which is both a quadratic field and a cyclotomic field (since i is a 4th root of unity). Like all quadratic fields it is a Galois extension of Q with Galois group cyclic of order two, in this case generated by complex conjugation, and is thus an abelian extension of Q, with conductor 4.^[1]

As with cyclotomic fields more generally, the field of Gaussian rationals is neither ordered nor complete (as a metric space). The Gaussian integers Z[i] form the ring of integers of Q(i). The set of all Gaussian rationals is countably infinite.

The field of Gaussian rationals is also a two-dimensional vector space over Q with natural basis ${\displaystyle \{1,i\}}$ .

The concept of Ford circles can be generalized from the rational numbers to the Gaussian rationals, giving Ford spheres. In this construction, the complex numbers are embedded as a plane in a three-dimensional Euclidean space, and for each Gaussian rational point in this plane one constructs a sphere tangent to the plane at that point. For a Gaussian rational represented in lowest terms as ${\displaystyle p/q}$ , the radius of this sphere should be ${\displaystyle 1/q{\bar {q}}}$  where ${\displaystyle {\bar {q}}}$  represents the complex conjugate of ${\displaystyle q}$ . The resulting spheres are tangent for pairs of Gaussian rationals ${\displaystyle P/Q}$  and ${\displaystyle p/q}$  with ${\displaystyle |Pq-pQ|=1}$ , and otherwise they do not intersect each other.^[2][3]