Chebyshev polynomials and Galois groups of De Moivre polynomials

Kurt Girstmair

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arXiv · 2007.14183

Chebyshev polynomials and Galois groups of De Moivre polynomials

Kurt Girstmair

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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.

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Kurt Girstmair. 2020-07-28. Chebyshev polynomials and Galois groups of De Moivre polynomials. https://arxiv.org/abs/2007.14183

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