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?

When taking quotient by $(t^2+1)$, the image of $t$ satisfies $t^2 + 1 = 0$, or, equivalently, $t^2 = -1$, so $t$ works as the imaginary unit.
Now, any element of the quotient can be uniquely represented by a polinomial with degree $\leq 1$ (using the fact that applying the identity $t^2 = -1$ reduces the degree). So, any element can be written as $a + tb$. Here, $a$ can be considered to be the real part and $b$ to be the imaginary part (just replace $t$ by $i$, if you feel better with that).
Addition of such polynomial is trivial (component-wise). Multiplication is a bit harder, but still very simple: $(a + bt)(c+dt) = ac + adt + bct + bdt^2 = (ac - db) + (ad + bc)t$ (since $t^2 = -1$). So, operations on this quotient ring are exactly those of complex numbers.
