  We mainly study a polynomial $f_{1,n}(x)=x^{n-1} + 2x^{n-2} + 3x^{n-3} +
\cdots + kx^{n-k} + \cdots + (n-1)x + n$ over $\mathbb{Z}$ and the Galois group
of the minimal splitting field. First, we show that an arbitrary root
$\alpha_{n}$ of $f_{1,n}(x)$ satisfies $|\alpha_{n}|\to 1$ ($n\to \infty$), and
discuss the irreducibility of $f_{1,n}(x)$ over $\mathbb{Z}$ for several type
$n$. After that, we show that the Galois group of $f_{1,n}(x)$ is Symmetric
group $S_{n-1}$ for several type $n$. Although those roots of $f_{1,n}(x)=0$
don't draw an exact circle, it looks like a circle on complex plane. Moreover
by considering that Galois groups of $f_{1,n}(x)$ are not abelian in many
cases, we call such extension fields over $\mathbb{Q}$ "Non-Abelian Cycrotomic
Fields" here.
