Factorization of univariate polynomials over some number field with Sympy

Question

I am working on factoring multivariate polynomials over some extension fields, using Sympy.

If I can factor univariate polynomials over the reals, I think I would have a working code. For my code this bring me down to factoring a univariate polynomial over 'QQ', and if needed, over some number field.

My approach now is to define these univariate polynomials over 'QQ', then look at the roots and decide for each root if it is real or not. If it is real, I add needed terms to 'QQ' and then ask Sympy to factor. This means that I try to automate the following steps:

1. f=Poly((x^2-3)*(x^2-5),x,domain='QQ')
2. solve(f,x)
3. (gives [-sqrt(3),sqrt(3),-sqrt(5),sqrt(5)])
4. f.factor(f,extension=[sqrt(3),sqrt(5)])

(..or some other way, but with similar steps and runtime I think)

This ofcourse has a very long runtime, since you are sort of twice calculating the factors. And there are also a lot of exceptions I need to think about as well.

Long story short: is there a way to ask Sympy to factor a polynomial over 'QQ' and allowing it to make some extensions if needed?

Is there something like f.factor(numberfield=True)?

Thank you in advance!!

Answer 1

This is planned, but not implemented yet (as of version 1.2). See Factoring polynomials into linear factors (emphasis mine):

Currently SymPy can factor polynomials into irreducibles over various domains, which can result in a splitting factorization (into linear factors). However, there is currently no systematic way to infer a splitting field (algebraic number field) automatically. In future the following syntax will be implemented:

factor(x**3 + x**2 - 7, split=True)

Note this is different from extension=True, because the later only tells how expression parsing should be done, not what should be the domain of computation. One can simulate the split keyword for several classes of polynomials using solve() function.

... where the last sentence refers to what you are doing now.