How do you prove the injective nature of an isomorphism from $\mathbb{R}[x]/\langle x^2+1\rangle$ and $\mathbb{C}$ I've been reviewing my notes for a course I'm taking and I am confused about my professor shows that $\mathbb{F}=\mathbb{R}[x]/\langle x^2+1\rangle$ is isomorphic to $\mathbb{C}$. I understand most of it but when it comes time to prove that the homomorphism is a bijection I get confused. Essentially, what he does is he shows that $f:\mathbb{F} \to \mathbb{C}: f(g)=g(i),g\in \mathbb{F}$ is a homomorphism and then shows the injective nature by showing that $\ker(f)={1_\mathbb{F} }$ and then he shows the bijective nature. 
I dont understand why this is a valid way of showing the injective nature of $f$. Could someone please explain it to me? Also, if you dont mind, would you be able to walk me through the logic of this proof?
Thanks

First it is worth to look into the elements of the field  $\mathbb R[x]/\langle x^2+1\rangle$.
Since $\mathbb R$ is a field $\mathbb R[x]$ is an Euclidean domain so the division algorithm holds.
hence any element of $\mathbb R[x]/\langle x^2+1\rangle$ is of the form $a+b\alpha $ where $\alpha $ satisfies $\alpha ^2+1=0$ and $a,b\in \mathbb R$.
Now consider the homomorphism $\phi:\mathbb R[x]/\langle x^2+1\rangle\to \mathbb C$ by $\phi(a+b\alpha)=a+bi$
