The irreduc function tests whether a multivariate polynomial over an algebraic number field is irreducible.  It returns true if a is irreducible, false otherwise.  Note that a constant polynomial by convention is reducible.

The call irreduc(a) tests for irreducibility over the field implied by the coefficients present; if all the coefficients are rational, then the irreducibility test is over the rationals.

The call irreduc(a, K) tests for irreducibility over the algebraic number field defined by K. K must be a single RootOf, a list or set of RootOfs, a single radical, or a list or set of radicals.

$\mathrm{irreduc}\left(2\right)$

$\mathrm{false}$

$\mathrm{irreduc}\left(x^3+5\right)$

$\mathrm{true}$

$\mathrm{irreduc}\left(x^3+5\,5^{\frac{1}{3}}\right)$

$\mathrm{false}$

$\mathrm{factor}\left(x^3+5\,5^{\frac{1}{3}}\right)$

$\left(5^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}x+x^2\right)\left(x+5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)$

$\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left(x^3-5\right)\right)\:$

$\mathrm{irreduc}\left(x^3+5\,\mathrm{\alpha }\right)$

$\mathrm{false}$

$\mathrm{factor}\left(x^3+5\,\mathrm{\alpha }\right)$

$\left({\mathrm{\alpha }}^2-\mathrm{\alpha }x+x^2\right)\left(x+\mathrm{\alpha }\right)$