Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On some determinants involving cyclotomic units   For each odd prime $p$, let $\zeta_p$ denote a primitive $p$-th root of
unity. In this paper, we study the determinants of some matrices with
cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and
$p>3$ the determinant of the matrix
$\(\frac{1-\zeta_p^{j^2k^2}}{1-\zeta_p^{j^2}}\)_{1\le j,k\le (p-1)/2}$ can be
written as $(-1)^{\frac{h(-p)+1}{2}}(a_p+b_pi\sqrt{p})$ with
$a_p,b_p\in\frac12\Z$ and
$$\begin{cases}\nu_p(a_p)=\nu_p(b_p)=\frac{p-3}{8}&\mbox{if}\ p\equiv 3\pmod8,
\\\nu_p(a_p)=\nu_p(b_p)+1=\frac{p+1}{8}&\mbox{if}\ p\equiv
7\pmod8,\end{cases}$$ where $\nu_p(x)$ denotes the $p$-adic order of a $p$-adic
integer $x$, and $h(-p)$ denotes the class number of the field $\Q(\sqrt{-p})$.
Meanwhile, let $(\frac{\cdot}{p})$ denote the Legendre symbol. We have
$$2^{\frac{p+1}{2}}a_pb_p=(-1){^\frac{p+1}{4}}p^{\frac{p-3}{4}}\det [S(p)],$$
and $$2^{\frac{p-1}{2}}(a_p^2-pb_p^2)=\frac{p-1}{2}(-p)^{\frac{p-3}{4}}\det
[S(p)],$$ where $\det [S(p)]$ is the determinant of the $\frac{p-1}{2}$ by
$\frac{p-1}{2}$ matrix $S(p)$ with entries $S(p)_{j,k}=(\frac{j^2+k^2}{p})$ for
any $1\le j,k\le (p-1)/2$.

6
The categories of this paper are: astro-ph
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
18
The categories of this paper are: econ
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg