Abstract In this paper, we investigate the p -adic valuation ν p (i (K) ) of the index i (K) of an algebraic number field K = Q (θ) K=Q (), where θ is a root of an irreducible sextinomial of the type x n + a x m + e x 3 + b x 2 + c x + d ∈ Z x x^n+ax^m+ex^3+bx^2+cx+d Zx. For a given rational prime p and certain natural numbers i p, we provide infinite families of number fields K for which ν p (i (K) ) = i p. In particular, i 2 ∈ 1, 2, 3, 4, 5, 6, 7, 8, while for every odd prime p, we construct families for which i p ∈ 1, 2, 3, p − 2, p − 1, p, p + 2. Several explicit examples are provided to illustrate the theoretical results.
Godara et al. (Tue,) studied this question.