Chebyshev polynomials and Galois groups of De Moivre polynomials

TL;DR

This paper investigates the Galois groups of De Moivre polynomials for all odd degrees, using Chebyshev polynomials to simplify the analysis and express zeros as rational functions of three key zeros.

Contribution

It provides a comprehensive description of Galois groups for arbitrary odd degrees and introduces a simplified Chebyshev polynomial approach for analyzing these polynomials.

Findings

Galois groups characterized for all odd degrees in the irreducible case

Zeros expressed as rational functions of three zeros

Simplified method based on Chebyshev polynomials

Abstract

Let $n\ge 3$ be an odd natural number. In 1738, Abraham de Moivre introduced a family of polynomials of degree $n$ with rational coefficients, all of which are solvable. So far, the Galois groups of these polynomials have been investigated only for prime numbers $n$ and under special assumptions. We describe the Galois groups for arbitrary odd $n\ge 3$ in the irreducible case, up to few exceptions. In addition, we express all zeros of such a polynomial as rational functions of three zeros, two of which are connected in a certain sense. These results are based on the reduction of the radical $$ \sqrt[n]{d+\sqrt R}, $$ whose degree is $2n$ in general, to irrationals of degree $\le n$. Such a reduction was given in a previous paper of the author. Here, however, we present a much simpler approach that is based on properties of Chebyshev polynomials.