TITLE: Polynomials for which roots can be expressed as polynomials in a single root
QUESTION [11 upvotes]: Classical Galois theory gives necessary and sufficient conditions for the roots of a polynomial in $k[x]$ to be expressible in terms of nested radicals of the coefficients.
Suppose instead that a single root $\alpha$ of $p(x)\in \mathbb{Q}[x]$ is known. Are there known necessary and sufficient conditions on $p(x)$ such that all remaining roots can be expressed as polynomial (or rational) functions of $\alpha$ and the coefficients of $p(x)$?
For example, the cyclotomic polynomials have this property, since every primitive $n^{\textrm{th}}$ root of unity can be written as a power of some fixed root.

REPLY [14 votes]: From a computational point of view, one should not try to compute the Galois group. Assuming $p(x) \in \mathbb{Q}[x]$ is irreducible, and $\alpha$ is a root of $p(x)$, it is sufficient to factor $p(x)$ over the number field $\mathbb{Q}(\alpha) = \mathbb{Q}[x]/(p(x))$, and look whether all the irreducible factors have degree 1. In this way, you also get the expression of the roots in terms of $\alpha$. This is much less expensive than computing the Galois group, which is feasible only in relatively low degree.