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

TL;DR

This paper investigates common divisors of values polynomials and their relation to indices in number fields, providing explicit computations for cubic and certain low-degree fields, and addressing open questions and conjectures in the area.

Contribution

It establishes the existence of number fields with prime divisors of their index for primes up to the degree, computes indices for specific fields, and refutes a previous theorem.

Findings

Existence of degree n fields with prime p dividing i(K) for p ≤ n

Explicit computation of i(K) for cubic fields

Counterexample to a previous theorem in the literature

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.