Singular Manual: factorize

  | ### 5.1.36 factorize `Syntax:` `factorize (` poly_expression `)` `factorize (` poly_expression`, 0 )` `factorize (` poly_expression`, 2 )` `Type:` list of ideal and intvec `Syntax:` `factorize (` poly_expression`, 1 )` `Type:` ideal `Purpose:` computes the irreducible factors (as an ideal) of the polynomial together with or without the multiplicities (as an intvec) depending on the second argument:
          0: returns factors and multiplicities, first factor is a constant.
             May also be written with only one argument.
          1: returns non-constant factors (no multiplicities).
          2: returns non-constant factors and multiplicities.
  `Note:` Not implemented for the coefficient fields real, finite fields of type `(p^n,a)` and ZZ/m. `Example:`
        ring r=32003,(x,y,z),dp;
        factorize(9*(x-1)^2*(y+z));
      ==> [1]:
      ==>    _[1]=9
      ==>    _[2]=y+z
      ==>    _[3]=x-1
      ==> [2]:
      ==>    1,1,2
        factorize(9*(x-1)^2*(y+z),1);
      ==> _[1]=y+z
      ==> _[2]=x-1
        factorize(9*(x-1)^2*(y+z),2);
      ==> [1]:
      ==>    _[1]=y+z
      ==>    _[2]=x-1
      ==> [2]:
      ==>    1,2
        ring rQ=0,x,dp;
        poly f = x2+1;            // irreducible in Q[x]
        factorize(f);
      ==> [1]:
      ==>    _[1]=1
      ==>    _[2]=x2+1
      ==> [2]:
      ==>    1,1
        ring rQi = (0,i),x,dp;
        minpoly = i2+1;
        poly f = x2+1;            // splits into linear factors in Q(i)[x]
        factorize(f);
      ==> [1]:
      ==>    _[1]=1
      ==>    _[2]=x+(-i)
      ==>    _[3]=x+(i)
      ==> [2]:
      ==>    1,1,1
See absFactorize; poly.

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