Polynomial Dedekind domains with finite residue fields of prime characteristic

We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb Z[X]$ and Q[X]$ with property its residue fields of prime characteristic are finite is equal to a generalized ring integer-valued polynomials, is, for each $p\in\mathbb Z$ there exists subset $E_p$ transcendental elements over Q$ in absolute integral closure $\overline{\mathbb Z_p}$ $p$-adic integers such $R=\{f\in\mathbb Q[X]\mid f(E_p)\subseteq \overline{\mathbb Z_p}, \forall \text{ }p\in\mathbb Z\}$. Moreover, we prove class group isomorphic direct sum countable family finitely generated abelian groups. Conversely, any this kind Q[X]$.