On non-monogenic number fields defined by trinomials of type $x^n +ax^m+b$

TL;DR

This paper investigates conditions under which number fields generated by roots of specific trinomials are not monogenic, providing explicit criteria and constructing infinite families of such non-monogenic fields.

Contribution

It offers explicit non-monogenity criteria for fields defined by trinomials and constructs infinite families of non-monogenic fields for degrees involving powers of 2 and 3.

Findings

Identifies explicit conditions for non-monogenity of certain trinomial-generated fields.

Constructs infinite families of non-monogenic fields of degrees 2^r*3^k.

Provides examples illustrating the theoretical results.

Abstract

Let $K=\Q(\theta)$ be a number field generated by a complex root $\th$ of a monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. In this paper, we deal with the problem of the non-monogenity of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As application, 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 $k$ are positive integers. We also give two infinite families of non-monogenic number fields defined by trinomials of degree $6$. Finally, we illustrate our results by giving some examples.