The NumberOfIrreduciblePolynomials(n, p, m) command computes the number of monic irreducible univariate polynomials of degree n over a finite field of order $p^m$.

An explicit formula for this function is $\frac{1}{n}\sum _{d|n}\mathrm{\mu }\left(d\right){\left(p^m\right)}^{\frac{n}{d}}$ where the sum is over the divisors of n and $\mathrm{\mu }$ is the Moebius function.

$\mathrm{with}\left(\mathrm{NumberTheory}\right)\:$

$\mathrm{NumberOfIrreduciblePolynomials}\left(3\,5\right)$

$40$

$\mathrm{NumberOfIrreduciblePolynomials}\left(1\,2^4\right)$

$16$

$\mathrm{seq}\left(\mathrm{NumberOfIrreduciblePolynomials}\left(n\,p\right)\,n=1..4\right)$

$p,\frac{1}{2}{p}^2-\frac{1}{2}p,\frac{1}{3}{p}^3-\frac{1}{3}p,\frac{1}{4}{p}^4-\frac{1}{4}{p}^2$

$\mathrm{NumberOfIrreduciblePolynomials}\left(3\,p\,m\right)$

$\frac{{\left(p^m\right)}^3}{3}-\frac{p^m}{3}$

The NumberTheory[NumberOfIrreduciblePolynomials] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.