Constructing infinite families of number fields with fixed indices via sextinomials

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 ) $K=\mathbb{Q}(\theta )$ , 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 ] ${x}^{n}+a{x}^{m}+e{x}^{3}+b{x}^{2}+cx+d\in \mathbb{Z}[x]$ . 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.