Is there an algorithm to check if a polynomial with integer coefficients is reducible over some extension of $\mathbb Q$?

Question

I have been reading some solved questions here, for example:

Showing a polynomial is irreducible over an extension field.

Is $f(y) = 5 - 4y^2 + y^4$ irreducible over $\mathbb{Q}(\sqrt{5})$?

showing how certain polynomials, of degree 3, 4 and 5, are irreducible over some respective extensions of $\mathbb Q$ which are proper in $\mathbb R$. But it seems that every answer uses a different method or approach, or that these polynomials have special features allowing them to be checked. Hence my question.

One can also ask how would you come up with a question like that? How to make sure that some polynomial can in fact be verified to be irreducible over some field $\mathbb Q(\alpha)$ for this irrational $\alpha$ ?

Answer 1

Yes, You can try looking for rational root test and Eisenstein’s criterion. I only know two such as Kronecker's method and the Zassenhaus algorithm but they only workout with full factorisation.

Answer 2

You could always use a factorization algorithm, as an inspiration I suggest to look at what computer algebra systems use. For example, PARI/GP has a function nffactor for factorization of univariate polynomial over given number field. Per documentation it implements Belabas's variant of van Hoeij's algorithm, so you can check the corresponding article A relative van Hoeij algorithm over number fields by Karim Belabas. I suspect other systems use same or similar algorithm.