Field Indices and Common Index Divisors of Generalized Eisenstein Trinomial Fields

Let [Formula: see text] be a prime number and let [Formula: see text], where [Formula: see text] is a root of the generalized Eisenstein trinomial [Formula: see text] with [Formula: see text] and [Formula: see text]. The arithmetic study of indices, common index divisors and monogenity constitutes a classical topic in algebraic number theory, dating back to the foundational works of Dedekind, Hensel, Engström and Ore. While numerous recent investigations have focused on monogenity and the computation of the index of a fixed generator, considerably less is known about the intrinsic field index and the common index divisor problem for infinite families of sparse polynomial number fields. The principal aim of this paper is to determine the field index [Formula: see text] and to completely resolve the common index divisor problem for the family of prime-degree generalized Eisenstein trinomial fields defined above. Using Dedekind’s criterion, Ore’s index theorem, Newton polygon techniques and the Hensel–Engström criterion, we prove that [Formula: see text] Consequently, the field [Formula: see text] possesses no common index divisors. To achieve this objective, we perform a detailed local analysis of all primes that may contribute to the index. For [Formula: see text], we derive an explicit formula for the index [Formula: see text] determine the precise contributions of the ramified primes to both the index and the discriminant, and obtain explicit arithmetic conditions ensuring monogenity. We further investigate the arithmetic consequences of the discriminant structure and establish a criterion for the determination of the Galois group in the special case [Formula: see text]. The results provide a new infinite family of prime-degree generalized Eisenstein trinomial fields for which the field index, the index of the defining generator and the common index divisor problem can be completely described. In particular, the paper complements and extends recent work on sparse polynomial fields by combining local index computations, Newton polygons, common index divisor theory and monogenity techniques within a unified arithmetic framework.