ON FIXED DIVISORS OF THE MINIMAL POLYNOMIALS OVER Z OF ALGEBRAIC NUMBERS

Abstract

International audienceLet 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 Q. For any γ ∈ K n , let F γ (x) be the unique irreducible polynomial in Z[x], such that its leading coefficient is positive and F γ (γ) = 0. Let i(γ) = gcd x∈Z F γ (x), i(K) = lcm θ ∈A n i(θ) andî(K) = lcm γ∈K n i(γ). For any γ ∈ K n , there exists a unique pair (θ , d), where θ ∈ A n and d is a positive integer such that γ = θ /d and θ ≡ 0 (mod p) for any prime divisor p of d. In this paper, among other results, we show that i(K) andî(K) have the same prime factors and we study the possible values of ν p (d) when p|i(γ). We introduce and study a new invariant of K defined using ν p (d), when γ describes K n. In the last theorem of this paper, we establish a generalisation of a theorem of MacCluer. From Lemma 2.1 we see that any prime factor p of c(γ) divides d(γ). Summarizing the relations between c(γ) and d(γ), we have : Remark 2.1. Let K be a number field of degree n and γ ∈ K n , then d(γ) and c(γ) have the same prime factors and for any prime p, we have ν p (d(γ)) ≤ ν p (c(γ)) ≤ nν p (d(γ))