Let $f(x) \in \mathbb{Z}[x]$ be a polynomial of degree $n$.
Question: Is there a nice way to describe the set of polynomials $f(x)$ (not necessarily monic) that can be factored as $f(x)= \prod_{i = 1}^n (\beta_i x - \alpha_i)$ over the splitting field in such a way that the factors are Galois conjugates of each other?
What I know: Evidently, if $f_0 = 1$, we can take $\beta_i = 1$ and the factors $(x - \alpha_i)$ are Galois conjugates of each other. Also, there are polynomials $f(x)$ that fail to satisfy this property, like $f(x) = 2x^2 - 5$.