Steps to construct the Field of fractions of Gaussian Integers $\mathbb{Z}[i]$

Question

i don't know how to construct such field $\mathbb{Q[i]} $ from $\mathbb{Z[i]}$. I know the following: $(a+bi,c+di)\sim (m+ni,r+si)$ iff $(a+bi)(r+si)=(c+di)(m+ni)$ is the equivalence relation and if $\frac ab+\frac cdi\in\Bbb Q(i)$, then: $$\frac ab+\frac cdi = \frac {ad+bci}{bd},$$ where both numerator and denominator are Gaussian Integers. On the other hand, I know that i need to use the universality of field of fractions, but I'm not clear how to construct such field.

Answer 1

There is no need to construct the field of fractions of $\mathbb{Z}[i]$ specifically, since the construction works "uniformly" for every integral domain.

The field of fractions of $\mathbb{Z}[i]$ is $\mathbb{Q}[i] \subseteq \mathbb{C}$ because $\mathbb{Q}[i]$ is a field containing $\mathbb{Z}[i]$, and clearly the smallest one.