On fixed divisors of the values of the minimal polynomials over Z of algebraic numbers

Abstract

Let $K$ be a number field of degree $n$, $A$ be its ring of integers, and $A_n$ (resp. $K_n$) be the set of elements of $A$ ( resp. $K$) which are primitive over $\mathbb Q$. For any $\gamma \in {K_n}$, let $F_{\gamma} (x)$ be the unique irreducible polynomial in $\mathbb Z[x]$, such that its leading coefficient is positive and $F_{\gamma} ({\gamma}) = 0$. Let $i(\gamma)=\gcd_{x\in\mathbb Z}F_{\gamma}(x)$, $i(K)=\lcm_{\theta\in{A_n}}i(\theta)$ and $\hat{\imath}(K) = \lcm_{\gamma\in{K_n}}i(\gamma)$. For any $\gamma \in {K_n}$, there exists a unique pair $(\theta,d)$, where $\theta\in A_n$ and $d$ is a positive integer such that $\gamma=\theta/d$ and $\theta\not\equiv 0\pmod{p}$ for any prime divisor $p$ of $d$. In this paper, we study the possible values of $\nu_{p}(d)$ when $p | i(\gamma)$. We introduce and study a new invariant of $K$ defined using $\nu_{p}(d)$, when $\gamma$ describes $K_n$. In the last theorem of this paper, we establish a generalisation of a theorem of MacCluer.