Let [Formula: see text] denote the ring of algebraic integers of an algebraic number field [Formula: see text], where [Formula: see text] is a root of an irreducible trinomial [Formula: see text] belonging to [Formula: see text]. In this paper, we give necessary and sufficient conditions involving only [Formula: see text] for a given prime [Formula: see text] to divide the index of the subgroup [Formula: see text] in [Formula: see text]. In particular, we deduce necessary and sufficient conditions for [Formula: see text] to be equal to [Formula: see text].

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a) be an extension of degree n of the field Q of rational numbers, where the integer a is such that for each prime p dividing n either p a or the highest power of p dividing a is coprime to p; this condition is clearly satisfied when a, n are coprime or a is squarefree. The paper contains an explicit formula for the discriminant of K involving only the prime powers dividing a, n.

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