The following passage from Wikipedia's article on quotient rings gives an example of a quotient ring of polynomials over a finite field.
"One important instance of the previous example is the construction of the finite fields. Consider for instance the field $F_3 = Z/3Z$ with three elements. The polynomial $f(x) = x^2 + 1$ is irreducible over $F_3$ (since it has no root), and we can construct the quotient ring $F_3[x]/(f)$. This is a field with $3^2=9$ elements, denoted by $F_9$. The other finite fields can be constructed in a similar fashion."
I have two questions: (1) Why must the resulting field have $3^2=9$ elements? (2) Why does $f(x)$ having no root imply that it's irreducible?