On prime divisors of the index of an algebraic integer

Let AKAK denote the ring of algebraic integers of an algebraic number field K=Q(θ)K=Q(θ) where the algebraic integer θ   has minimal polynomial F(x)=xn+axm+bF(x)=xn+axm+b over the field QQ of rational numbers with n=mt+un=mt+u, t∈Nt∈N, 0≤u≤m−10≤u≤m−1. In this paper, we characterize those primes which divide the discriminant of F(x)F(x) but do not divide [AK:Z[θ]][AK:Z[θ]] when u=0u=0 or u divides m; such primes p   are important for explicitly determining the decomposition of pAKpAK into a product of prime ideals of AKAK in view of the well known Dedekind theorem. As a consequence, we obtain some necessary and sufficient conditions involving only a, b, m, n   for AKAK to be equal to Z[θ]Z[θ].