On index divisors and monogenity of certain number fields defined by $x^{12}+ax^m+b$

TL;DR

This paper investigates the conditions under which certain 12th-degree number fields, defined by specific trinomials, are not monogenic, providing criteria related to prime divisors of the index and partial solutions to a classical problem.

Contribution

It offers new sufficient conditions for non-monogenity of fields defined by $x^{12}+ax^m+b$, especially for $m=1$, and characterizes prime divisors of the index, advancing understanding of monogenity in these fields.

Findings

Criteria for non-monogenity based on prime divisors

Characterization of primes dividing the index for $m=1$

Partial solution to Narkiewicz's Problem 22

Abstract

In this paper, we deal with the problem of monogenity of number fields defined by monic irreducible trinomials $F(x)=x^{12}+ax^m+b\in \mathbb{Z}[x]$ with $1\leq m\leq11$. We give sufficient conditions on $a$, $b$, and $m$ so that the number field $K$ is not monogenic. In particular, for $m=1$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$ and we provide a partial answer to the Problem $22$ of Narkiewicz \cite{Nar} for these number fields. Our results are illustrated by computational examples.