Quadratic residues and related permutations

TL;DR

This paper investigates permutations related to quadratic residues modulo an odd prime p, determining their signs and exploring properties of sequences formed by quadratic residues and primitive roots.

Contribution

It provides new results on the signs of specific permutations involving quadratic residues, extending previous work by explicitly calculating these signs for different congruence classes of p.

Findings

Sign of permutation σ_{0,1} for p ≡ 3 mod 4 determined.

Sign of permutation σ_{0,2} for p ≡ 1 mod 4 established.

Properties of sequences involving quadratic residues and primitive roots analyzed.

Abstract

Let $p$ be an odd prime. For any $p$-adic integer $a$ we let $\overline{a}$ denote the unique integer $x$ with $-p/2<x<p/2$ and $x-a$ divisible by $p$. In this paper we study some permutations involving quadratic residues modulo $p$. For instance, we consider the following three sequences. \begin{align*} &A_0: \overline{1^2},\ \overline{2^2},\ \cdots,\ \overline{((p-1)/2)^2},\\ &A_1: \overline{a_1},\ \overline{a_2},\ \cdots,\ \overline{a_{(p-1)/2}},\\ &A_2: \overline{g^2},\ \overline{g^4},\ \cdots,\ \overline{g^{p-1}}, \end{align*} where $g\in\Z$ 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 $\sigma_{0,1}$ and determine the sign $\sigma_{0,2}$ when $p\equiv 1\pmod 4$.

Equations

$$A_0:\overline{1^2},\overline{2^2},\cdots ,\overline{((p-1)/2{)}^2},$$

$$A_1:\overline{a_1},\overline{a_2},\cdots ,\overline{a_{(p-1)/2}},$$

$$A_2:\overline{g^2},\overline{g^4},\cdots ,\overline{g^{p-1}},$$

$$\prod _{1\le i<j\le \frac{p-1}{2}}({\zeta }_p^{j^2}-{\zeta }_p^{i^2}),$$

$$A_0:\overline{1^2},\overline{2^2},\cdots ,\overline{((p-1)/2{)}^2},$$

$$A_1:\overline{a_1},\overline{a_2},\cdots ,\overline{a_{(p-1)/2}},$$

$$A_2:\overline{g^2},\overline{g^4},\cdots ,\overline{g^{p-1}},$$

$$\mathrm{sgn}({\sigma }_{0,1})=\{\begin{array}{ll}1& \text{mbox}ifp\equiv 3(\mathrm{mod}8),\\ (-1{)}^{(h(-p)+1)/2}& \text{mbox}ifp\equiv 7(\mathrm{mod}8),\end{array}$$

$$s_p:=\prod _{\begin{array}{c}1\le k\le (p-1)/2\\ (\frac{k}{p})=-1\end{array}}k$$

$$r_{p}^{*}:=\#\{(x,y):1\leq x,y\leq(p-1)/2,\ x+y\leq(p-1)/2,\ \bigg{(}\frac{x}{p}\bigg{)}=\bigg{(}\frac{y}{p}\bigg{)}=1\},$$

$$\mathrm{sgn}({\sigma }_{0,1})=\{\begin{array}{ll}\overline{s_p{u}_p^{(p-1)/4}}\cdot (-1{)}^{((h(p)+2)(p-1)+2h(-4p))/8+r_p^{\ast }}& \text{mbox}ifp\equiv 1(\mathrm{mod}8),\\ & \\ \overline{\frac{1}{2}{s}_p{u}_p^{(p+3)/4}}\cdot (-1{)}^{((h(p)+2)(p+3)+2h(-4p)-4)/8+r_p^{\ast }}& \text{mbox}ifp\equiv 5(\mathrm{mod}8).\end{array}$$

$$\mathrm{sgn}({\sigma }_{0,2})=\overline{\frac{1}{2}{n}^{n/2}{u}_p{g}^{(p-1)(3n^2-n-2)/8}}\cdot (-1{)}^{(h(p)+1)/2}.$$

$$\mathrm{sgn}({\sigma }_{0,2})=\overline{\frac{1}{2}{n}^{n/2}{u}_p}\cdot (-1{)}^{(2h(p)+n)/4}.$$

$$M_p=\left(\begin{array}{cccc}\chi (1^2)& \chi (2^2)& \cdots & \chi ((\frac{p-1}{2}{)}^2)\\ {\chi }^2(1^2)& {\chi }^2(2^2)& \cdots & {\chi }^2((\frac{p-1}{2}{)}^2)\\ ⋮& ⋮& \ddots & ⋮\\ {\chi }^{\frac{p-1}{2}}(1^2)& {\chi }^{\frac{p-1}{2}}(2^2)& \cdots & {\chi }^{\frac{p-1}{2}}((\frac{p-1}{2}{)}^2)\end{array}\right).$$

$$\det(M_{p})=n^{n/2}(-1)^{(n+2)/4}\times\bigg{(}\overline{\frac{1}{2}n^{n/2}u_{p}}\cdot(-1)^{(2h(p)+n)/4}\bigg{)}\in\mathbb{Z}.$$

$$\prod _{1\le i<j\le n}(j^2-i^2)\equiv \{\begin{array}{ll}-n!(\mathrm{mod}p)& \text{mbox}ifp\equiv 1(\mathrm{mod}4),\\ 1(\mathrm{mod}p)& \text{mbox}ifp\equiv 3(\mathrm{mod}4).\end{array}$$

$$\bigg{(}\frac{p-1}{2}\bigg{)}!\equiv(-1)^{\frac{h(-p)+1}{2}}\ ({\rm{mod}}\ p),$$

$$\bigg{(}\frac{p-1}{2}\bigg{)}!\equiv\frac{(-1)^{(h(p)+1)/2}u_{p}}{2}\ ({\rm{mod}}\ p),$$

$$2^{\frac{p-1}{4}}\equiv \{\begin{array}{ll}(-1{)}^{(p-1)/8+h(-4p)/4}(\mathrm{mod}p)& \text{mbox}ifp\equiv 1(\mathrm{mod}8),\\ (-1{)}^{(p-5)/8+(h(-4p)-2))/4}\cdot (\frac{p-1}{2})!(\mathrm{mod}p)& \text{mbox}ifp\equiv 5(\mathrm{mod}8).\end{array}$$

$$\mathrm{sgn}({\sigma }_{0,1})=\prod _{1\le i<j\le n}\frac{\overline{j^2}-\overline{i^2}}{\overline{a_j}-\overline{a_i}}.$$

$$\prod _{1\le i<j\le n}(j^2-i^2)\equiv -n!\equiv -1\cdot \frac{(-1{)}^{(h(p)+1)/2}{u}_p}{2}(\mathrm{mod}p).$$

$$\prod _{1\le i<j\le n}(a_j-a_i)=\prod _{1\le k\le p-1}{k}^{A_k^{++}},$$

$$A_k^{--}:=$$

$$A_k^{+-}:=$$

$$A_k^{-+}:=$$

$$A_k^{++}+A_k^{+-}=r_k,\text{and}{A}_k^{-+}+A_k^{--}=p-1-k-r_k.$$

$$A_k^{+-}=A_k^{-+}.$$

$$A_k^{++}+A_k^{+-}+A_k^{-+}+A_k^{--}=p-1-k,$$

$$A_k^{+-}+A_k^{--}=p-1-k-r_k.$$

$$A_{k}^{++}+A_{k}^{--}-A_{k}^{+-}-A_{k}^{-+}=\sum_{1\leq x\leq p-1-k}\bigg{(}\frac{x}{p}\bigg{)}\bigg{(}\frac{x+k}{p}\bigg{)}.$$

$$A_{k}^{++}=r_{k}+\frac{-p+k+1}{4}+\frac{1}{4}\sum_{1\leq x\leq p-1-k}\bigg{(}\frac{x^{2}+kx}{p}\bigg{)}.$$

$$A_{p-k}^{++}=r_{p-k}+\frac{-k+1}{4}+\frac{1}{4}\sum_{1\leq x\leq k-1}\bigg{(}\frac{x^{2}-kx}{p}\bigg{)}.$$

$$\displaystyle r_{p-k}=\#\{1\leq x\leq k-1:\bigg{(}\frac{x}{p}\bigg{)}=1\}=\#\{p+1-k\leq x\leq p-1:\bigg{(}\frac{x}{p}\bigg{)}=1\},$$

Full text

[TABLE]

where $g\in\mathbb{Z}$ is a primitive root modulo $p$ and $1\leq a_{1}<a_{2}<\cdots<a_{(p-1)/2}\leq p-1$ 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 $p\equiv 3\ ({\rm{mod}}\ 4)$. In this paper we give the sign of $\sigma_{0,1}$ and determine the sign $\sigma_{0,2}$ when $p\equiv 1\ ({\rm{mod}}\ 4)$.

2010 Mathematics Subject Classification. Primary 11A15; Secondary 05A05, 11R18.

Keywords. quadratic residues, permutations, primitive roots, cyclotomic fields.

Supported by the National Natural Science Foundation of China (Grant No. 11571162).

1. Introduction

Let $S$ be a finite set, and let $\tau$ be a permutation on $S$. Throughout this paper the sign of $\tau$ is denoted by ${\rm sgn}(\tau)$. Investigating the properties of $\tau$ is a classical topic in number theory and combinatorics. In particular, when $S$ is the finite field $\mathbb{F}_{p}$ with $p>2$ prime and $\tau$ is induced by a permutation polynomial $f(x)$ over $\mathbb{F}_{p}$, the properties of $\tau$ have been studied deeply by many mathematicians.

For instance, when $f(x)=ax$ with $a\in\mathbb{Z}$ and $\gcd(a,p)=1$, the famous Zolotarev’s Theorem [10] states that the Legendre symbol $(\frac{a}{p})$ is the sign of the permutation of $\mathbb{Z}/p\mathbb{Z}$ induced by multiplication by $a$. Using this, Zolotarev obtained a new proof of the law of quadratic reciprocity. In this line, Sun [6] investigated many permutation problems involving quadratic permutation polynomial over $\mathbb{F}_{p}$. When $k\in\mathbb{Z}^{+}$ and $\gcd(k,p-1)=1$, it is easy to see that $f(x)=x^{k}$ is a permutation polynomial over $\mathbb{F}_{p}$. L.-Y Wang and the author [8] determined the sign of permutation $\tau$ induced by $f(x)$. Moreover, W. Duke and K. Hopkins [2] extended this topic to an arbitrary finite group. They also generalized the law of quadratic reciprocity on finite groups.

Given an odd prime $p$. Throughout this paper, for any integer $a$ we let $\overline{a}$ denote the unique integer $x$ with $-p/2<x<p/2$ and $x-a$ divisible by $p$. Let $1=a_{1}<a_{2}<\cdots<a_{\frac{p-1}{2}}\leq p-1$ be all quadratic residues modulo $p$. Note that $\overline{1^{2}},\ \overline{2^{2}},\ \cdots,\ \overline{((p-1)/2)^{2}}$ is a permutation of $\overline{a_{1}},\ \overline{a_{2}},\ \cdots,\ \overline{a_{(p-1)/2}}$. Sun first studied this problem and obtained the sign of this permutation in the case of $p\equiv 3\ ({\rm{mod}}\ 4)$ by evaluating the product

[TABLE]

where $\zeta_{p}=e^{2\pi i/p}$. Inspired by Sun’s work, we consider the following sequences.

[TABLE]

where $g\in\mathbb{Z}$ is a primitive root modulo $p$. Clearly $A_{i}$ is a permutation of $A_{j}$ and we call this permutation $\sigma_{i,j}$. As mentioned before, when $p\equiv 3\ ({\rm{mod}}\ 4)$, Sun proved that

[TABLE]

where $h(-p)$ is the class number of $\mathbb{Q}(\sqrt{-p})$.

In this paper, we study the above permutations. We first introduce some notations. Let $p\equiv 1\ ({\rm{mod}}\ 4)$ be a prime. Let $\varepsilon_{p}=(u_{p}+v_{p}\sqrt{p})/2>1$ and $h(p)$ be the fundamental unit and the class number of $\mathbb{Q}(\sqrt{p})$ respectively. And we let $h(-4p)$ denote the class number of $\mathbb{Q}(\sqrt{-p})$. We also define

[TABLE]

and let

[TABLE]

where $\#S$ denotes the cardinality of a finite set $S$. Now we are in the position to state our first theorem.

Theorem 1.1**.**

Let $p\equiv 1\ ({\rm{mod}}\ 4)$ be a prime. Then

[TABLE]

Now we turn to the permutation $\sigma_{0,2}$. For convenience, we write $p=2n+1$ for some positive integer $n$.

Theorem 1.2**.**

Let $p\equiv 1\ ({\rm{mod}}\ 4)$ be a prime, and let $g\in\mathbb{Z}$ be a primitive root modulo $p$ Then

[TABLE]

In particular, when $p\equiv 5\ ({\rm{mod}}\ 8)$, ${\rm sgn}(\sigma_{0,2})$ is independent on the choice of $g$ and we have

[TABLE]

As an application of Theorem 1.2, we can calculate some determinants concerning Dirichlet characters modulo $p$. Let $p$ be an odd prime and let $g$ be a primitive root modulo $p$. Let $\widehat{\mathbb{F}_{p}}$ denote the group of Dirichlet characters modulo $p$ and let $\chi$ be a generator of $\widehat{\mathbb{F}_{p}}$. We consider the following matrix:

[TABLE]

We have the following result.

Corollary 1.1**.**

Let $p\equiv 5\ ({\rm{mod}}\ 8)$ be a prime. Then

[TABLE]

Remark 1.1*.*

The signs of permutations have deep connections with the calculations of determinants. The readers may see [7] for more examples.

The proofs of Theorem 1.1–1.2 and Corollary 1.1 will be given in the next section.

2. Proofs of Theorem 1.1–1.2 and Corollary 1.1

Let $p$ be an odd prime. Throughout this section, we set $p=2n+1$. We begin with the following lemma (cf. [6, (1.5)]).

Lemma 2.1**.**

[TABLE]

Mordell [4] showed that if $p>3$ is a prime and $p\equiv 3\ ({\rm{mod}}\ 4)$ then

[TABLE]

where $h(-p)$ is the class number of $\mathbb{Q}(\sqrt{-p})$. Later S. Chowla [1] extended Mordell’s result and obtained the following result.

Lemma 2.2**.**

Let $p\equiv 1\ ({\rm{mod}}\ 4)$ be a prime. Then we have

[TABLE]

where $h(p)$ and $u_{p}$ are defined as in Theorem 1.1.

In 1982 K. S. Williams and J. D. Currie [9] obtained the following result.

Lemma 2.3**.**

Let $p\equiv 1\ ({\rm{mod}}\ 4)$ be a prime. We have

[TABLE]

Proof of Theorem 1.1. Let $p\equiv 1\ ({\rm{mod}}\ 4)$ be a prime. It follows from definition that

[TABLE]

We first consider the numerator. By Lemma 2.1–2.2 we have

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We now turn to the denominator. It is clear that

[TABLE]

where $A_{k}^{++}:=\#\{1\leq x\leq p-1-k:(\frac{x}{p})=(\frac{x+k}{p})=1\}$. To calculate $A_{k}^{++}$, we set

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We also let $r_{k}:=\#\{1\leq x\leq p-1-k:(\frac{x}{p})=1\}.$ Then clearly we have

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From the map $x\mapsto p-k-x$, we immediately obtain that

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Hence $A_{k}^{++}+A_{k}^{-+}=r_{k}.$ Noting that

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we obtain that

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Moreover, we have

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Combining the above equations, we obtain

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Replacing $k$ by $p-k$, we get

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Noting that

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we therefore have

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Observing that

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we obtain

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The last equation follows from [3, p.63 Exercise 8]. In view of the above, we have

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Thus

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Clearly

[TABLE]

where $r_{p}^{*}$ is defined as in Theorem 1.1. We also have the following identities.

[TABLE]

Hence we get

[TABLE]

Combining (2.1) and (2.2), our result follows from Lemma 2.2–2.3. ∎

Now we concentrate on Theorem 1.2. Let $\zeta=e^{2\pi i/(p-1)}$ . We obtain the following lemma.

Lemma 2.4**.**

Let $p>3$ be a prime. We have the identity

[TABLE]

In particular, if $p\equiv 5\ ({\rm{mod}}\ 8)$, then the product is equal to $n^{\frac{n}{2}}(-1)^{\frac{n-2}{4}}\in\mathbb{Z}$.

Proof.

Let $f(\zeta)=\prod_{1\leq i<j\leq n}(\zeta^{2j}-\zeta^{2i})$. Then

[TABLE]

Hence $\mid f(\zeta)\mid=n^{n/2}$. Next we consider the argument ${\rm Arg}(f(\zeta))$ of $f(\zeta)$. Since

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we have

[TABLE]

Hence

[TABLE]

Thus $f(\zeta)=n^{n/2}e^{(3n^{2}-n-2)\pi i/4}$. If $p\equiv 5\ ({\rm{mod}}\ 8)$, then $n\equiv 2\ ({\rm{mod}}\ 4)$. Noting that $\frac{3n^{2}-n-2}{4}\in\mathbb{Z}$ and $\frac{3n^{2}-n-2}{4}\equiv\frac{n-2}{4}\ ({\rm{mod}}\ 2)$, we have $f(\zeta)=n^{n/2}(-1)^{(n-2)/4}\in\mathbb{Z}.$ ∎

Remark 2.1*.*

Let $\chi$ and $g$ be as in Corollary 1.1. Let

[TABLE]

If $\chi(g)=\zeta$, then

[TABLE]

Proof of Theorem 1.2. Let $p\equiv 1\ ({\rm{mod}}\ 4)$ be a prime, and let $g\in\mathbb{Z}$ be a primitive root modulo $p$. By definition we have

[TABLE]

As in the proof of Theorem 1.1, the numerator

[TABLE]

We mainly focus on the denominator. We first observe the following fact. Let $\Phi_{p-1}(x)$ be the $(p-1)$-th cyclotomic polynomial. Since $p>2$ and $p\equiv 1\ ({\rm{mod}}\ p-1)$, it is known that $p$ totally splits in $\mathbb{Q}(\zeta)$. Hence by Kummer’s Theorem (cf. [5, p.47 Proposition 8.3]) we know that $\Phi_{p-1}(x)\ ({\rm{mod}}\ p\mathbb{Z}[x])$ splits in $\mathbb{Z}/p\mathbb{Z}[x]$. And the set of primitive $(p-1)$-th roots of unity of $\mathbb{Q}(\zeta)$ maps bijectively onto the set of primitive $(p-1)$-th roots of unity of $\mathbb{Z}/p\mathbb{Z}$. Hence we have

[TABLE]

Now let $f(x)=\prod_{1\leq i<j\leq n}(x^{2j}-x^{2i})$. We may write

[TABLE]

with $g(x),h(x)\in\mathbb{Z}[x]$ and $\deg(h(x))<\deg(\Phi_{p-1}(x))$. By Lemma 2.4 we know that

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As $p\equiv 1\ ({\rm{mod}}\ 4)$, we have $n^{\frac{n}{2}}\in\mathbb{Z}$. Thus by Galois theory for each primitive $(p-1)$-th root of unity $\zeta_{p-1}$ we have

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Let

[TABLE]

We obtain

[TABLE]

Hence we have

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In particular, in view of the above, when $p\equiv 5\ ({\rm{mod}}\ 8)$, for each primitive $(p-1)$-th root of unity $\zeta_{p-1}$ we have

[TABLE]

Hence $h(x)$ is the constant $n^{\frac{n}{2}}\cdot(-1)^{\frac{n-2}{4}}$. The desired result follows from (2.3) and (2.4).∎

Proof of Corollary 1.1. Let $M_{p}$ be as in Corollary 1.1, and let $N_{p,g}$ be as in Remark 2.1. When $p\equiv 5\ ({\rm{mod}}\ 8)$, it is easy to see that $\det(N_{p,g})$ is independent on the choice of $g$. Then

[TABLE]

Our desired result follows from Lemma 2.4 and Theorem 1.2.∎

Acknowledgments We are exceedingly thankful for the careful reading and indispensable suggestions of the anonymous referees. We also thank Prof. Z.-W Sun and Prof. Hao Pan for their help suggestions.

This research was supported by the National Natural Science Foundation of China (Grant No. 11571162).