I could use some help with the following exercise:
Find the number of reducible monic polynomials of degree $2$ over $\mathbb F_p$. Show this implies that for every prime $p$ there exists a field of order $p^2$.
I found the number of reducible polynomials of degree $2$ is $\frac{p(p+1)}{2}$ since a reducible polynomial of degree $2$ can be written as the product $(x-a)(x-b)$ for $a,b\in \mathbb F_p$.
Since there are $p^2$ polynomials of degree $2$ over $\mathbb F_p$, there are $\frac{p^2-p}{2}$ irreducible polynomials of degree $2$ over $\mathbb F_p$. We can choose an irreducible polynomial $f$ as such, and so $\mathbb F_p / (f)$ is a field (since $(f)$ is a maximal ideal because $f$ is irreducible). But why does $\mathbb F_p / (f)$ have $p^2$ elements?
