On indices and monogenity of quartic number fields defined by quadrinomials
Hamid BEN YAKKOU
Received April 11, 2025. Published online March 9, 2026.
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 $K$. H. T. Engstrom (1930) established that $i(K)=2^u \cdot3^v$ with $u \le2$ and $v \le1$. This implies that $i(K) \in\{1, 2, 3, 4, 6, 12\}$. 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 $i(K) \neq1$, $K$ cannot be monogenic. Moreover, we give infinite families of such quartic number fields whose indices belong to the set $\{1, 2, 3, 4, 6, 12\}$. We also identify new families of monogenic quartic number fields generated by roots of non-monogenic quadrinomials. Our method is based on a theorem of Ø. Ore (1928) on the decomposition of primes in number fields.
Keywords: quartic number field; power integral basis; monogenity; index of a number field; common index divisor; Newton polygon; theorem of Ore; prime ideal factorization
Affiliations: Hamid Ben Yakkou, LIMATI Laboratory, Polydisciplinary Faculty of Beni-Mellal, University Sultan Moulay Slimane, Beni-Mellal, Morocco, e-mail: {\tt\Brokenlinktext{beyakou}{hamid}{gmail.com}}, hamid.benyakkou@usms.ma
