Chebyshev polynomials and Galois groups of De Moivre polynomials

Abstract

Let [Formula: see text] 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 [Formula: see text] and under special assumptions. We describe the Galois groups for arbitrary odd numbers [Formula: see text] 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 an irrational of degree 2n to irrationals of degree [Formula: see text]. 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. We also give a simple proof of a result of Filaseta et al.