### 5.1.34 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 and finite fields of type `(p^n,a)`.
`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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