Common Divisors of Values Polynomials and common factors of indices in a Number Field

Abstract

Let $\mathbb{K}$ be a number field of degree $n$ over $\mathbb{Q}$. Let $\widehat{\mathbb{A}}$ be the set of integers of $\mathbb{K}$ which are primitive over $\mathbb{Q}$ and $I(\mathbb{K})$ be its index. Gunji and McQuillan defined the following integer $i(\mathbb{K})=\underset{\theta\in \widehat{\mathbb{A}}}{\text{lcm}}\;i(\theta)$, where $i(\theta)=\underset{x\in \mathbb{Z}}{\text{gcd}}\;F_\theta(x)$ and $F_\theta(x)$ is the characteristic polynomial of $\theta$ over $\mathbb{Q}$. We prove that if $p$ is a prime number less than or equal to $n$ then there exists a number field $\mathbb{K}$ of degree $n$ for which $p$ divides $i(\mathbb{K})$. We compute $i(\mathbb{K})$ for cubic fields. Also we determine $I(\mathbb{K})$ and $i(\mathbb{K})$ for families of simplest number fields of degree less than $7$. We give also answers to questions one and two in \cite{Kihel}. Furthermore, we give a counter example to Theorem 11 in \cite{Kihel} and we discuss their conjecture.