Prime Polynomial Theorem

Standard

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 $\mathbb{F}_q$ and of degree $n \sim \frac{q^n}{n}$, for a prime power $q$.

The proof of this theorem follows from Gauss’ formula:

# monic irreducible polynomialswith coefficients in $\mathbb{F}_q$ and of degree $n$ = $\displaystyle{\frac{1}{n}\sum_{d|n}\mu\left(\frac{n}{d}\right)q^d}$, by taking $d=n$.

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