Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Approach to Non-Abelian Cyclotomic Fields   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.

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The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: cond-mat