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?

The complex number $a+bi$ corresponds to the equivalence class of polynomials that contains the polynomial $bt+a$.
For example $2+3i$ corresponds to the equivalence class
$$ \{3t+2, t^2+3t+3, -t^2+3t+1, t^3+4t+2, \pi t^4+\pi t^2+3t+2,\ldots \} $$
You should be able to check that addition and multiplication of polynomials correspond to addition and multiplication of complex numbers -- that is, if $p(t)$ and $q(t)$ are polynomials that represent the complex numbers $z_1$ and $z_2$, then the polynomial $p(t)+q(t)$ represents the complex number $z_1+z_2$, and the polynomial $p(t)q(t)$ represents the complex number $z_1z_2$.
