# Quadratic residues and related permutations concerning cyclotomic fields

Research paper by **Hai-Liang Wu**

Indexed on: **05 Mar '19**Published on: **04 Mar '19**Published in: **arXiv - Mathematics - Number Theory**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### Abstract

Let $p$ be an odd prime. For any $p$-adic integer $a$ we let $a\ (p)$ denote
the unique integer $x$ with $-p/2<x<p/2$ and $x-a$ divisible by $p$. In this
article we first study some permutations involving quadratic residues modulo
$p$. For instance, we consider the following three sequences. \begin{align*}
&A_0: 1^2\ (p),\ 2^2\ (p),\ \cdots,\ ((p-1)/2)^2\ (p),\\ &A_1: a_1\ (p),\ a_2\
(p),\ \cdots,\ a_{(p-1)/2}\ (p),\\ &A_2: g^2\ (p),\ g^4\ (p),\ \cdots,\
g^{p-1}\ (p), \end{align*} where $g\in\mathbb{Z}$ is a primitive root modulo
$p$ and $1\le a_1<a_2<\cdots<a_{(p-1)/2}\le p-1$ are all quadratic residues
modulo $p$. Obviously $A_i$ is a permutation of $A_j$ and we call this
permutation $\sigma_{i,j}$. Sun obtained the sign of $\sigma_{0,1}$ when
$p\equiv 3\pmod4$. In this paper we give the sign of $\sigma_{0,1}$ when
$p\equiv 1\pmod 4$ and determine the sign $\sigma_{0,2}$ when $p\equiv
5\pmod8$. Moreover, we also investigate problems analogous to the above in the
ring of Gaussian integers. In the second part of this paper, let $n>2$ be a
positive integers. $\mathbb{Q}(\zeta_n)$ denotes the $n$-th cyclotomic field,
where $\zeta_n$ is a primitive $n$-th root of unity. Let
$p\mathbb{Z}[\zeta_n]=(\mathfrak{p}_1\cdots\mathfrak{p}_g)^e$ be the
factorization of $p\mathbb{Z}[\zeta_n]$ into prime ideals of
$\mathbb{Z}[\zeta_n]$. Then each element $\tau$ of the Galois group
$\Gal(\Q(\zeta_n)/\Q)$ induces a permutation $\tilde{\tau}$ on
$\mathfrak{p}_1,\ \cdots,\ \mathfrak{p}_g$ by sending $\mathfrak{p}_i$ to
$\tau(\mathfrak{p}_i)$. We study the structure and the sign of $\tilde{\tau}$
in this paper.