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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 $\alpha$ ?

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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.

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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.

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  • $\begingroup$ The first two do not provide an algorithm for testing irreducibility. If you find a rational root, and the polynomial is of degree greater than 1, then it is reducible. And if you don't... then what? If you can find a prime $p$ such that the polynomial is Eisenstein at $p$, then it is irreducible. And if you don't find such a prime... then what? $\endgroup$ Feb 5 at 1:09

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