5 research outputs found
On prime divisors of the index of an algebraic integer
Let A<sub>K</sub> denote the ring of algebraic integers of an algebraic number field K = Q(θ) where the algebraic integer θ has minimal polynomial F(x) = x<sup>n</sup> + ax<sup>m</sup> + 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<sub>K</sub> : Z[θ]] when u = 0 or u divides m; such primes p are important for explicitly determining the decomposition of pA<sub>K</sub> into a product of prime ideals of A<sub>K</sub> 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<sub>K</sub> to be equal to Z[θ]