# Factoring bivariate polynomials w.r.t. a single variable

the function **factor()** works in this fashion:

```
sage: x, y = PolynomialRing(GF(17), 2, ['x','y']).gens()
sage: f = 9*y^6 - 9*x^2*y^5 - 18*x^3*y^4 - 9*x^5*y^4 + 9*x^6*y^2 + 9*x^7*y^3 + 18*x^8*y^2 - 9*x^11
sage: f.factor()
(-9) * (x^5 - y^2) * (x^6 - 2*x^3*y^2 - x^2*y^3 + y^4)
```

Is there a possibility to factorize a bivariate polynomial in x,y with respect to a single variable only (e.g. y) and get the answer in the form of **(y - f(x))** as factors?

Surely there is no essential difference between the factorization in $F[x,y]$ which you see and the one in $F(x)[y]$ which you want, by Gauss's Lemma? The fact that none of the displayed factors is linear in $y$ just means that $f$ has no roots in $F(x)$ when you consider it as a polynomial in $F(x)[y]$. To get roots (and hence linear factors) you would need to enlarge $F(x)$ to its algebraic closure.