On prime divisors of the index of an algebraic integer

Abstract

Let A_K denote the ring of algebraic integers of an algebraic number field K = Q(θ) where the algebraic integer θ has minimal polynomial F(x) = xⁿ + ax^m + b over the field Q of 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 [A_K : Z[θ]] when u = 0 or u divides m; such primes p are important for explicitly determining the decomposition of pA_K into a product of prime ideals of A_K 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 A_K to be equal to Z[θ].