1. ## proving an isomorphism

How do i prove that the set of all complex numbers of the form a + bi with a, b in Q (rationals) is isomorphic to the field of fractions of the ring Z[i] of Gaussian integers?

2. Originally Posted by Louise
looking at them as subsets of $\displaystyle \mathbb{C},$ they're more than just isomorphic. they are basically equal. because an element of the field of fractions of $\displaystyle \mathbb{Z}[i]$ is in the form $\displaystyle x=\frac{a+bi}{c+di},$ where $\displaystyle a,b,c,d \in \mathbb{Z}$
and $\displaystyle cd \neq 0.$ but we have $\displaystyle x=\frac{(a+bi)(c-di)}{c^2+d^2}=\frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i \in \mathbb{Q}(i).$ on the other hand if $\displaystyle y \in \mathbb{Q}(i),$ then $\displaystyle y=p+qi,$ for some $\displaystyle p=\frac{a}{b}, \ q=\frac{c}{d},$ with $\displaystyle a,b,c,d \in \mathbb{Z},$ and $\displaystyle bd \neq 0.$ then
$\displaystyle y=\frac{ad + bc i}{bd}.$ clearly both $\displaystyle ad + bc i$ and $\displaystyle bd$ are in $\displaystyle \mathbb{Z}[i]$ and thus $\displaystyle y$ is in the field of fractions of $\displaystyle \mathbb{Z}[i]$ because $\displaystyle bd \neq 0.$