On indices and monogenity of quartic number fields defined by quadrinomials

TL;DR

This paper investigates the divisibility of the index of quartic number fields generated by specific quadrinomials, providing conditions for divisibility by 2 or 3, and identifies new families of monogenic fields.

Contribution

It offers new criteria for the divisibility of the index in quartic fields and constructs infinite families of monogenic fields from non-monogenic quadrinomials.

Findings

Conditions for divisibility of index by 2 and 3.

Identification of infinite families of monogenic quartic fields.

Application of Ore's theorem for prime decomposition.

Abstract

Consider a quartic number field $K$ generated by a root of an irreducible quadrinomial of the form $ F(x)= x^4+ax^3+bx+c \in \Z[x]$. Let $i(K)$ denote the index of $K$. Engstrom \cite{Engstrom} established that $i(K)=2^u \cdot 3^v$ with $u \le 2$ and $v \le 1$. 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 $i(K) \neq 1$, $K$ 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 \cite{Nar,O}.