Set of roots Hermite polynomials(probabilistic type)

Question

What is the nature of the set of all roots of the Hermite polynomials? They’re known to be all real. Is it a dense set? If not, what are the limit points?Are the limit points also roots? Are the limit points also roots(of course, the limit point 0 is a root)? If not, are they also algebric integers, or even algebraic? (Since these polynomials are all essentially monic, they’re algebraic integers. The even Hermite polynomials that I have checked so far satisfy Eisenstein’s Criterion, and the odd polynomials that I have checked essentially satisfy the criterion after factoring out x. I haven’t proved that all Hermite polynomials satisfy or essentially satisfy it, but it would be nice to know that if all of the polynomials were either irreducible or essentially so after factoring out x.)