Decomposition of primes in non-maximal orders

Let R be a Dedekind ring with quotient field K, and let F (x) ∈ R[x] be an irreducible primitive polynomial. Let α be a root of F and L = K(α). If F is monic, then R[α] is a monogenic R-order of L. If F is not monic, then one can construct an R-order RF of L that suitably generalizes R[α]. We show that, even if F is not monic, the classical invariants of RF can be derived directly from the polynomial F , precisely as in the case of monogenic orders.