On prime divisors of the index of an algebraic integer

Abstract

Let AKdenote the ring of algebraic integers of an algebraic number field K=Q(θ)where the algebraic integer θhas minimal polynomial F(x) =xn+axm+bover the field Qof rational numbers with n =mt +u, t ∈N, 0 ≤u ≤m −1. In this paper, we characterize those primes which divide the discriminant of F(x) but do not divide [AK:Z[θ]] when u =0or udivides m; such primes pare important for explicitly determining the decomposition of pAKinto a product of prime ideals of AKin view of the well known Dedekind theorem. As a consequence, we obtain some necessary and sufficient conditions involving only a, b, m, nfor AKto be equal to Z[θ]