On index divisors and monogenity of certain number fields defined by trinomials

TL;DR

This paper investigates conditions under which certain number fields generated by roots of specific trinomials are not monogenic, providing explicit criteria and constructing infinite families of such non-monogenic fields, especially for degrees involving powers of 2 and 3.

Contribution

It offers explicit conditions for non-monogenity in number fields defined by trinomials and constructs infinite families of non-monogenic fields for degrees like 2^r·3^k and sextic fields.

Findings

Identifies explicit conditions for non-monogenity.

Constructs infinite families of non-monogenic fields.

Provides examples illustrating the theoretical results.

Abstract

Let $K$ be a number field generated by a root $\th$ of a monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. In this paper, we study the problem of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree $n=2^r\cdot3^k$ with $r$ and $k$ two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this paper.