numtheory/mipolys(deprecated) - Help

numtheory

 mipolys
 number of monic irreducible univariate polynomials

 Calling Sequence mipolys(n, p, m)

Parameters

 n - non-negative integer p - prime integer (characteristic of a finite field) m - (optional) positive integer

Description

 • Important: The numtheory[mipolys] command 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 $Z\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}p$, 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

$\left(\frac{1}{n}\right)\left(\sum \phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\mathrm{mobius}\left(\frac{n}{d}\right){\left({p}^{m}\right)}^{d},\mathrm{for}d\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∈\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{divisors}\left(n\right)\right)$

Examples

 > $\mathrm{with}\left(\mathrm{numtheory}\right):$
 > $\mathrm{mipolys}\left(3,5\right)$
 ${40}$ (1)
 > $\mathrm{mipolys}\left(1,2,4\right)$
 ${16}$ (2)
 > $\mathrm{seq}\left(\mathrm{mipolys}\left(n,p\right),n=1..4\right)$
 ${p}{,}\frac{{1}}{{2}}{}{{p}}^{{2}}{-}\frac{{1}}{{2}}{}{p}{,}\frac{{1}}{{3}}{}{{p}}^{{3}}{-}\frac{{1}}{{3}}{}{p}{,}\frac{{1}}{{4}}{}{{p}}^{{4}}{-}\frac{{1}}{{4}}{}{{p}}^{{2}}$ (3)
 > $\mathrm{mipolys}\left(3,p,4\right)$
 $\frac{{1}}{{3}}{}{{p}}^{{12}}{-}\frac{{1}}{{3}}{}{{p}}^{{4}}$ (4)
 > $\mathrm{mipolys}\left(3,p,k\right)$
 $\frac{{1}}{{3}}{}{\left({{p}}^{{k}}\right)}^{{3}}{-}\frac{{1}}{{3}}{}{{p}}^{{k}}$ (5)