Q:

Proving that this mapping is one to one

Let $Q$ be the field of quotients of the Gaussian integers (integer complex numbers) and let $R$ be the the set of all complex numbers of the form $a +bi$ such that both $a,b$ are rationals
I have to prove that the mapping $\phi:Q \longrightarrow R$ such that $ \phi ([m+ni,r+si]) = \frac{m+ni}{r+si}$ I already proved that $\phi$ preserves the ring structure and i also proved that it is onto. I have to prove now that it is one to one to conclude that $\phi$ is a ring isomoprhim. Here is my attempt
assume $\phi([m_1+n_1i,r_1+s_1i])=\phi([m_2+n_2i,r_2+s_2i])$ then this implies that $\frac{m_1+n_1i}{r_1+s_1i}=\frac{m_2+n_2i}{r_2+s_2i}$ but now where to go from here , any suggetsions

A:

Remember that to show injectivity, it sufficies to show that if $\phi(x) = 0$, then $x=0$. So, what equation do you get when you write $\phi([m+ni,r+si]) = 0$?

