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M Ayad ; A Bayad ; O Kihel - On fixed divisors of the values of the minimal polynomials over Z of algebraic numbers

hrj:10748 - Hardy-Ramanujan Journal, February 6, 2023, Volume 45 - 2022 - https://doi.org/10.46298/hrj.2023.10748
On fixed divisors of the values of the minimal polynomials over Z of algebraic numbersArticle

Authors: M Ayad 1; A Bayad ; O Kihel

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 γKn, 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)=gcdxZFγ(x), i(K)=\lcmθAni(θ) and ˆı(K)=\lcmγKni(γ). For any γKn, there exists a unique pair (θ,d), where θAn and d is a positive integer such that γ=θ/d and θ0(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.


Volume: Volume 45 - 2022
Published on: February 6, 2023
Imported on: December 28, 2022
Keywords: Values of polynomials,Denominators of algebraic numbers,Splitting of prime numbers,11R04, 12Y05,[MATH]Mathematics [math]
Funding:
    Source : OpenAIRE Graph
  • Funder: Natural Sciences and Engineering Research Council of Canada

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