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I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a lower bound on the degrees of the factors, by saying: there is a factor of size at least... ?

I know the result which says that $x^n-a$ is irreducible if $a$ is not in $\mathbb{Q}^p$ for some $p$ that divides $n$ and if we assume, in the case where $4|n$, that $a\not\in -4\mathbb{Q}^4$. [Lang, Algebra, VI, §9, Theorem 9.1] What happens in the other cases?

PS: This is related to another question I asked, which has still no answer: Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

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  • $\begingroup$ Davenport, J. H., Factorisation of sparse polynomials, Computer algebra (London, 1983), 214–224, Lecture Notes in Comput. Sci., 162, Springer, Berlin, 1983, MR0774813 (86i:11073) studies the factorization of these polynomials. $\endgroup$
    – Gerry Myerson
    Jul 23, 2014 at 6:07

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