On indices and monogenity of quartic number fields defined by quadrinomials

Consider a quartic number field $K$ generated by a root of an irreducible quadrinomial of the form $F(x)=x^4+a{x}^3+bx+c\in \mathbb{Z}[x]$. 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 $i(K)\mathrm{\ne }1$, $K$ 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}.