Let $\displaystyle K$ be a field and $\displaystyle f \in K[x]$. Prove that $\displaystyle K[x]/ \langle f \rangle$ is a field if and only if $\displaystyle f$ is irreductible.

How can I build a field with nine elements?

Prove that $\displaystyle R[x]/ \langle x^2 + 1 \rangle \cong \mathbf{C}$

Thanks!