Definition:Gaussian Rational/Definition 2

Definition
The field $\sqbrk {\Q \sqbrk i, +, \times}$ of Gaussian rationals is the field of quotients of the integral domain $\struct {\Z \sqbrk i, +, \times}$ of Gaussian integers.

This is shown to exist in Existence of Field of Quotients.

In view of Field of Quotients is Unique, we construct the field of quotients of $\Z \sqbrk i$, give it a label $\Q \sqbrk i$ and call its elements Gaussian rationals.

Also see

 * Equivalence of Definitions of Gaussian Rational