Constructing $\mathbb{C}$ from $\mathbb{R}$ I'm having difficulty grasping the notion that you can define the complex numbers as $\mathbb{C}=\mathbb{R}[t]/\langle t^2+1\rangle$.
As far as I understand, $\mathbb{R}[t]$ is the set of all polynomials in $t$ whose coefficients are real. 
When we take the quotient, we are forming equivalence classes. But how does (some polynomial in $[t]$ with real coefficients) + $(t^2+1)$ translate into a complex number? Or should the operation be $\times$ instead of $+$?
At that point, do you get the set of complex numbers by considering the roots of these polynomials?

There are two ways to see this:
1) The first is to give an isomorphism from $$\frac{\mathbb{R}[t]}{<t^2+1>} \rightarrow \mathbb{C}.$$
Define $\phi : \frac{\mathbb{R}[t]}{<t^2+1>} \rightarrow \mathbb{C}$ by
$\phi(r) = r$ for $r \in \mathbb{R}$ and $\phi(t) = i.$ 
We can easily verify that this gives an isomorphism between the two required set.
2) Another way to see this fact is to understand the meaning of quotients.
When we go quotient by an ideal, say $<t^2 + 1>$ then what we are doing is we are basically allowing the polynomial $t^2 +1 $ to have a solution in the quotient. This is essentially why we construct $\mathbb{C}$ in the first place. So the quotient is a ring which contains $\mathbb{R}$ and allows us to solve $t^2+1.$ And since $t^2+1$ is irreducible in $\mathbb{R}$ we see that $\frac{\mathbb{R}[t]}{<t^2+1>}$ is a field isomorphic to $\mathbb{C}.$
