Let $f(x) = ax^2+bx+c \in \Bbb{R}[x]$ be an irreducible polynomial. Prove that the field $\Bbb{R}[x] \mathbin{/} \langle f(x) \rangle$ is isomorphic to the field of complex numbers.
Knowing that $$ \Bbb{R}[x] \mathbin{/} \langle f(x) \rangle = \{h + gv \mid h, g \in \Bbb{R};\, av^2+bv+c = 0 \}, $$ how can I detect an element $\alpha \in\Bbb{R}[x] \mathbin{/} \langle f(x) \rangle$ such that $\alpha^2 = -1$?