I have two questions about the class of integer-coefficient polynomials all of whose roots are rational. I asked this at MSE, but it attracted little interest (perhaps because it is not interesting!)

Q1. Is there some way to recognize such a polynomial from its coefficients $a_0, a_1, \ldots, a_n$?

I am aware of the rational-root theorem, which says that each rational root is of the form $\pm p/q$, where $p$ is a factor of $a_0$ and $q$ a factor of $a_n$.

Example. The roots of $$ 12544 x^5 + 24976 x^4 - 23994 x^3 - 51721 x^2 - 17080 x + 1275 $$ are $$\lbrace \frac{3}{2}, -\frac{5}{7}, -\frac{5}{7}, \frac{1}{16}, -\frac{17}{8} \rbrace \;. $$ Here $a_0 = 1275 = 3 \cdot 5 \cdot 17$ and $a_5 = 12544 = 2^8 \cdot 7^2$.

As Mark Bennet commented at MSE, perhaps an analog of Sturm's theorem would serve.

Q2. Has this class of polynomials been studied in its own right?

In other words, is this class interesting? I can see it has at least a monoid structure, as the product of two such polynomials also has all rational roots.

These are naive questions, well out of my expertise. Thanks in advance for educating me!