The AIrreduc function is a placeholder for testing the absolute irreducibility of the polynomial P, that is irreducibility over an algebraic closure of its coefficient field. It is used in conjunction with evala.

The call evala(AIrreduc(P)) tests the absolute irreducibility of the polynomial P over the field of complex numbers. The polynomial P must have algebraic number coefficients in RootOf notation (see algnum).

A univariate polynomial is absolutely irreducible if and only if it is of degree 1.

The function AIrreduc looks for sufficient conditions of absolute reducibility or irreducibility. It returns true if the polynomial P is detected absolutely irreducible, false if it is detected absolutely reducible, FAIL otherwise.

In the case of nonrational coefficients, only trivial conditions are tested.

If the polynomial P has rational coefficients, an absolute irreducibility criterion is sought over the reduction of P modulo p, where p runs through a set of prime integers. If S is given, the primes in S are used. Otherwise, the first ten odd primes and the first five primes greater than the degree of P are chosen. Although the probability for P to be absolutely reducible in case of failure is not controlled, it is very likely that P can be factored.

$f≔92x^{9}y-93x^2{y}^2+91x^7{y}^3+y^4+x^{10}$

$f≔x^{10}+92x^{9}y+91x^7{y}^3-93x^2{y}^2+y^4$

$\mathrm{evala}\left(\mathrm{AIrreduc}\left(f\right)\right)$

$\mathrm{true}$

$\mathrm{alias}\left(a=\mathrm{RootOf}\left(T^3-2\right)\right)$

$a$

$f≔a{y}^3+a{y}^{3}x+13a{y}^2-7xy-7x^{2}y-91x$

$f≔a{y}^{3}x+a{y}^3+13a{y}^2-7x^{2}y-7xy-91x$

$\mathrm{evala}\left(\mathrm{AIrreduc}\left(f\right)\right)$

$\mathrm{false}$

$\mathrm{evala}\left(\mathrm{AFactor}\left(f\right)\right)$

$-7\left(xy+y+13\right)\left(-\frac{a{y}^2}{7}+x\right)$

$f≔-2y^2{x}^2+y^2+10x^{2}y+25x^4$

$f≔25x^4-2y^2{x}^2+10x^{2}y+y^2$

$\mathrm{evala}\left(\mathrm{AIrreduc}\left(f\right)\right)$

$\mathrm{FAIL}$

$\mathrm{evala}\left(\mathrm{AFactor}\left(f\right)\right)$

$25\left(\frac{xy\mathrm{RootOf}\left({\mathrm{\_Z}}^2-2\right)}{5}+x^2+\frac{y}{5}\right)\left(-\frac{xy\mathrm{RootOf}\left({\mathrm{\_Z}}^2-2\right)}{5}+x^2+\frac{y}{5}\right)$

$f≔48778x^3+275894451y^6-1188761805y^{5}x+1707361425y^4{x}^2-817400375x^3{y}^3+1232777y^3$

$f≔-817400375x^3{y}^3+1707361425y^4{x}^2-1188761805y^{5}x+275894451y^6+48778x^3+1232777y^3$

$\mathrm{evala}\left(\mathrm{AIrreduc}\left(f\right)\right)$

$\mathrm{FAIL}$

$\mathrm{evala}\left(\mathrm{AIrreduc}\left(f\,\left\{37\,43\right\}\right)\right)$

$\mathrm{true}$

Ragot, Jean-Francois. "Probabilistic Absolute Irreducibility Test of Polynomials." In Proceedings of MEGA '98. 1998.