Show that $\mathbb{R}[T]/(T-1)\equiv \mathbb{R}$ and $\mathbb{R}[T]/(T^2+1)\equiv \mathbb{C}$

Question

We have to find a surjective ring homomorphism $\mathbb{R}[T]\rightarrow \mathbb{R}$ whose kernel is $(T-1)$. I suspect that the map that sends $T$ to $1$ is the desired map. But what can I do to find the second map?

Answer 1

As hinted at in the comments, send $T$ to $1$ and $\pm i$ respectively. Of course you still have to check that it's well-defined and an isomorphism. More generally, for a field $k$ you can send $T \in k[x] /P(T) $ to a root of $P$ to get an isomorphism with $k$ adjoined that root.