  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.
