A short note on number fields defined by exponential Taylor polynomials

2023/03/10 by Anuj Jakhar, Jakhar, Anuj, Srinivas Kotyada +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2303.08100

openalex publication_date 2023/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let n be a positive integer and f_n(x)= 1+x+(x²)/(2!)+⋯ + (xⁿ)/(n!) denote the n-th Taylor polynomial of the exponential function. Let K = Q(θ) be an algebraic number field where θ is a root of f_n(x) and Z_K denote the ring of algebraic integers of K. In this paper, we prove that for any prime p, p does not divide the index of the subgroup Z[θ] in Z_K if and only if p²\nmid n!.

Related

• Algebraic properties of summation of exponential Taylor polynomials
• On power basis of a class of number fields
• A theorem of complete reducibility for exponential polynomials
• E-ideals in exponential polynomial ring
• On the arithmetic of polynomials over a number field