I just wanted to point towards a nice theorem, analogous to the Prime Number Theorem, which is not talked about much:

# irreducible monic polynomials with coefficients in and of degree , for a prime power .

The proof of this theorem follows from Gauss’ formula:

# monic irreducible polynomialswith coefficients in and of degree = , by taking .

For details, see first section of this: http://alpha.math.uga.edu/~pollack/thesis/thesis-final.pdf