I'm studying Artin's Algebra, and the question says to "Classify quadratic extensions of $\mathbb Q$." What would that look like?
A quadratic extension of $\mathbb Q$ is just, for $d$ square-free, $\{a + b\sqrt{d} : a, b \in\mathbb {Q} \}$, right? Asking to classify implies that two different values of $d$ can give the same extension?
If $\mathbb{Q}/(f_1)$ and $\mathbb{Q}/(f_2)$ are quadratic extensions of $\mathbb{Q}$ then the roots of $f_1$ and $f_2$ are the same (thus the extensions are the same) if $\Delta_1 = \left(u_1 - u_2\right)^2 = \Delta_2 = \left(v_1 - v_2\right)^2$ where $u_1, u_2$ are roots of $f_1$ and $v_1, v_2$ are roots of $f_2$.