An Approach to Non-Abelian Cyclotomic Fields

2017/01/04 by Shinji Ishida, Ishida, Shinji
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1701.01160

openalex publication_date 2017/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We mainly study a polynomial f_1,n(x)=x^n-1 + 2x^n-2 + 3x^n-3 + ⋯ + kx^n-k + ⋯ + (n-1)x + n over ℤ and the Galois group of the minimal splitting field. First, we show that an arbitrary root α_n of f_1,n(x) satisfies |α_n|→ 1 (n→ ∞), and discuss the irreducibility of f_1,n(x) over ℤ for several type n. After that, we show that the Galois group of f_1,n(x) is Symmetric group S_n-1 for several type n. Although those roots of f_1,n(x)=0 don't draw an exact circle, it looks like a circle on complex plane. Moreover by considering that Galois groups of f_1,n(x) are not abelian in many cases, we call such extension fields over ℚ "Non-Abelian Cycrotomic Fields" here.

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