切比雪夫多项式与De Moivre多项式的伽罗瓦群
Chebyshev polynomials and Galois groups of De Moivre polynomials
摘要：
设$n\ge 3$为奇数自然数。在1738年，亚伯拉罕·德·莫夫尔引入了一系列具有有理系数的$n$次多项式，所有这些多项式都是可解的。到目前为止，这些多项式的伽罗瓦群只研究了素数$n$和特殊假设。我们描述的伽罗瓦群的任意奇数$n\ge 3$在不可约的情况下，到少数例外。此外，我们还将这样一个多项式的所有零点表示为三个零点的有理函数，其中两个零点在某种意义上是连通的。这些结果是基于自由基$$ \sqrt[n]{d+\sqrt R}, $$的程度一般是$2n$，减少到有理数的程度$\le n$。作者在以前的一篇论文中给出了这种减少。然而，在这里，我们提出了一个更简单的方法，是基于切比雪夫多项式的属性。
英文摘要：
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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