Hardy-Ramanujan Journal |
Let K be a number field of degree n, A be its ring of integers, and An (resp. Kn) be the set of elements of A ( resp. K) which are primitive over Q. For any γ∈Kn, let Fγ(x) be the unique irreducible polynomial in Z[x], such that its leading coefficient is positive and Fγ(γ)=0. Let i(γ)=gcdx∈ZFγ(x), i(K)=\lcmθ∈Ani(θ) and ˆı(K)=\lcmγ∈Kni(γ). For any γ∈Kn, there exists a unique pair (θ,d), where θ∈An and d is a positive integer such that γ=θ/d and θ≢0(modp) for any prime divisor p of d. In this paper, we study the possible values of νp(d) when p|i(γ). We introduce and study a new invariant of K defined using νp(d), when γ describes Kn. In the last theorem of this paper, we establish a generalisation of a theorem of MacCluer.