On indices and monogenity of quartic number fields defined by quadrinomials

2024/01/23 by Hamid Ben Yakkou, Yakkou, Hamid Ben
Mathematics · #Algebraic Geometry and Number Theory #Finite Group Theory Research #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2401.12782

Abstract

Consider a quartic number field K generated by a root of an irreducible quadrinomial of the form F(x)= x⁴+ax³+bx+c ∈ \Z[x]. Let i(K) denote the index of K. Engstrom \citeEngstrom established that i(K)=2^u ⋅ 3^v with u ≤ 2 and v ≤ 1. In this paper, we provide sufficient conditions on a, b and c for i(K) to be divisible by 2 or 3, determining the exact corresponding values of u and v in each case. In particular, when i(K) ≠ 1, K cannot be monogenic. We also identify new infinite parametric families of monogenic quartic number fields generated by roots of non-monogenic quadrinomials. We illustrate our results by some computational examples. Our method is based on a theorem of Ore on the decomposition of primes in number fields \citeNar,O.

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