Let p be a prime. Determine the number of irreducible quadratic polynomials over Z_p
The polynomial factors into a product of monic irreducible polynomials of order dividing .
There are linear monic polynomials. Let be the number of monic irreducible quadradic polynomials.
Then by counting degrees of polynomials in
We see that .
(This formula can be generalized to polynomials of degree by applying Mobius inversion formula)