On common index divisors and monogenity of of the nonic number field defined by a trinomial $x^9+ax+b$

TL;DR

This paper investigates the conditions under which primes divide the index of nonic number fields generated by trinomials, providing criteria based on coefficients and identifying infinite families of non-monogenic fields.

Contribution

It offers necessary and sufficient conditions for prime divisors of the index in nonic fields defined by trinomials, and identifies infinite families of non-monogenic fields.

Findings

Criteria for prime divisors of the index based on coefficients a and b

Identification of infinite families of non-monogenic nonic fields

Numerical examples illustrating theoretical results

Abstract

Let $K $ be a nonic number field generated by a complex root $\th$ of a monic irreducible trinomial $ F(x)= x^9+ax+b \in \Z[x]$, where $ab \neq 0$. Let $i(K)$ be the index of $K$. A rational prime $p$ dividing $ i(K)$ is called a prime common index divisor of $K$. In this paper, for every rational prime $p$, we give necessary and sufficient conditions depending only $a$ and $b$ for which $p$ is a common index divisor of $K$. As application of our results we identify infinite parametric families of non-monogenic nonic numbers fields defined by such trinomials. At the end, some numerical examples illustrating our theoretical results are given.