The minimal periodicity for integral bases of pure number fields

2025/09/11 by Khai-Hoan Nguyen-Dang, Nguyen-Dang, Khai-Hoan · 1 citation
Mathematics · #11R04 #11R21 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.09457

openalex publication_date 2025/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Fix n≥3. For the pure field K_a=\mathbb Q(θ) with θⁿ=a, where a≠ ± 1 is nth-power-free, we encode an integral basis in the fixed coordinate \1,θ,…,θ^n-1\ by its shape. We prove a sharp local-to-global principle: for each p^e ∥ n, the local shape at p is determined by a\bmod p ^e+1, and this precision is optimal. Moreover, the global shape is periodic with minimal modulus M(n)=∏_{p^e∥ n}p ^e+1=n\cdotrad(n), providing many applications in the understanding integral bases of pure number fields.

Citations

• Moments of the distribution of k-free numbers in short intervals and arithmetic progressions
• On integral basis of pure number fields
• The Monogeneity of Kummer Extensions and Radical Extensions
• Complexity of OM factorizations of polynomials over local fields
• Higher Newton polygons and integral bases
• Newton polygons of higher order in algebraic number theory

Cited by

• α-monogeneity of pure number fields: criterion and density

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