An Approach to Non-Abelian Cyclotomic Fields

TL;DR

This paper investigates a family of polynomials with roots approaching the unit circle, explores their irreducibility, and shows their Galois groups are symmetric, leading to the concept of non-abelian cyclotomic fields.

Contribution

It introduces non-abelian cyclotomic fields by analyzing the roots and Galois groups of specific polynomials, extending classical cyclotomic field theory.

Findings

Roots tend to the unit circle as degree increases

Galois groups are often symmetric groups

Polynomials are irreducible for several types

Abstract

We mainly study a polynomial $f_{1,n}(x)=x^{n-1} + 2x^{n-2} + 3x^{n-3} + \cdots + kx^{n-k} + \cdots + (n-1)x + n$ over $\mathbb{Z}$ and the Galois group of the minimal splitting field. First, we show that an arbitrary root $\alpha_{n}$ of $f_{1,n}(x)$ satisfies $|\alpha_{n}|\to 1$ ($n\to \infty$), and discuss the irreducibility of $f_{1,n}(x)$ over $\mathbb{Z}$ 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 $\mathbb{Q}$ "Non-Abelian Cycrotomic Fields" here.