On common index divisors and monogenity of certain number fields defined by trinomials of type $x^{2^r}+ax^m+b$

TL;DR

This paper investigates the monogenity of number fields generated by roots of specific trinomials using $p$-adic Newton polygons, establishing conditions under which these fields are not monogenic and providing explicit examples.

Contribution

It introduces new criteria for non-monogenity of fields defined by trinomials of type $x^{2^r}+ax^m+b$, including explicit conditions and constructions of monogenic fields from non-monogenic ones.

Findings

If $a$ and $1+b$ are divisible by 32, the field is not monogenic.

Explicit conditions for non-monogenity when $m=1$.

Construction of monogenic fields from non-monogenic roots.

Abstract

Let $K = \Q(\th)$ be a number with $\th$ a root of an irreducible trinomial of type $ F(x)= x^{2^r}+ax^m+b \in \Z[x]$. In this paper, based on the $p$-adic Newton polygon techniques applied on decomposition of primes in number fields and the classical index theorem of Ore \cite{Narprime, O}, we study the monogenity of $K$. More precisely, we prove that if $a$ and $1+b$ are both divisible by $32$, then $K$ cannot be monogenic. For $m=1$, we provide explicit conditions on $a$, $b$ and $r$ for which $K$ is not monogenic. We also construct a family of irreducible trinomials which are not monogenic, but their roots generate monogenic number fields. To illustrate our results, we give some computational examples.