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irreduc

polynomial irreducibility test

 Calling Sequence irreduc(a) irreduc(a, K)

Parameters

 a - multivariate polynomial K - (optional) algebraic number field extension

Description

 • 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.

Examples

 > $\mathrm{irreduc}\left(2\right)$
 ${\mathrm{false}}$ (1)
 > $\mathrm{irreduc}\left({x}^{3}+5\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{irreduc}\left({x}^{3}+5,{5}^{\frac{1}{3}}\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{factor}\left({x}^{3}+5,{5}^{\frac{1}{3}}\right)$
 $\left({{5}}^{{2}}{{3}}}{-}{{5}}^{{1}}{{3}}}{}{x}{+}{{x}}^{{2}}\right){}\left({x}{+}{{5}}^{{1}}{{3}}}\right)$ (4)
 > $\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({x}^{3}-5\right)\right):$
 > $\mathrm{irreduc}\left({x}^{3}+5,\mathrm{\alpha }\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{factor}\left({x}^{3}+5,\mathrm{\alpha }\right)$
 $\left({{\mathrm{\alpha }}}^{{2}}{-}{\mathrm{\alpha }}{}{x}{+}{{x}}^{{2}}\right){}\left({x}{+}{\mathrm{\alpha }}\right)$ (6)