论文标题
Chebyshev多项式和de Moivre多项式的Galois组
Chebyshev polynomials and Galois groups of De Moivre polynomials
论文作者
论文摘要
令$ n \ ge 3 $为奇怪的数字。 1738年，亚伯拉罕·德·莫伊夫（Abraham de Moivre）引入了一个多项式$ n $的多项式家庭，所有这些系数均可解决。到目前为止，这些多项式的Galois组仅针对质量数$ n $和特殊假设进行了研究。我们在不可约的情况下描述了Galois组的任意奇数$ n \ ge 3 $，最多例外。此外，我们表达了这样一个多项式的所有零，例如三个零的合理函数，其中两个是在某种意义上连接的。这些结果基于降低激进$$ \ sqrt [n] {d+\ sqrt r}，$$的$ $ $ \ le n $。作者的先前论文中给出了这样的减少。但是，在这里，我们提出了一种基于Chebyshev多项式的属性的简单方法。
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.
