Prove that $\mathbb Q[i]$ is the field of fractions of $\mathbb Z[i]$.

Question

I need to prove that $\mathbb Q[i]$ is the field of fractions of $\mathbb Z[i]$. Can someone help me?

Answer 1

If you know your theorems about field of fractions, you can prove it using the universal properties for fields of fractions. Alternatively, you know that $\mathbb{Q}[x]/\langle x^2+1 \rangle$ is a field (because $x^2+1$) is irreducible. Can you find a map $$ \phi: \mathbb{Q}[i] \to \mathbb{Q}[x]/\langle x^2+1 \rangle $$ and prove it is an isomorphism? [Hint: What should $i$ map to?]