Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[NumberOfIrreduciblePolynomials] instead.

The mipolys function computes the number of monic irreducible univariate polynomials of degree n over the finite field $Z\mathbf{mod}p$, if the parameter m is not specified.

If m is specified, mipolys(n, p, m) computes the number of monic irreducible univariate polynomials of degree n over the Galois field $\mathrm{GF}\left(p^m\right)$.

If m is not explicitly specified, m defaults to 1. In this context, the general mathematical definition of mipolys is

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

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

$40$

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

$16$

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

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

$\mathrm{mipolys}\left(3\,p\,4\right)$

$\frac{1}{3}{p}^{12}-\frac{1}{3}{p}^4$

$\mathrm{mipolys}\left(3\,p\,k\right)$

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