I have two questions about the class of integer-coefficient polynomials all of whose roots are rational.
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 $a_n$.
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!
Update. The discussion at MathOverflow (see here) concluded that Q1 can be answered in polynomial time, using methods in the famous Lenstra, Lenstra, Lovász paper, "Factoring polynomials with rational coefficients."