## 25 Citations

Determinants concerning Legendre symbols

- MathematicsComptes Rendus. Mathématique
- 2021

The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove…

Quadratic residues and related permutations and identities

- MathematicsFinite Fields Their Appl.
- 2019

On some determinants involving the tangent function

- Mathematics
- 2019

Let $p$ be an odd prime and let $a,b\in\mathbb Z$ with $p\nmid ab$. In this paper we mainly evaluate $$T_p^{(\delta)}(a,b):=\det\left[\tan\pi\frac{aj^2+bk^2}p\right]_{\delta\le j,k\le (p-1)/2}\ \…

On some determinants involving cyclotomic units

- Mathematics
- 2019

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…

Elliptic curves over finite fields and determinants with Legendre symbol entries.

- Mathematics
- 2020

Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In…

Sums and products of quadratic residues and related identities

- Mathematics
- 2020

In this paper we study some sums and products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $…

Quadratic residues and quartic residues modulo primes

- Mathematics
- 2018

In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product…

Quadratic residues and related permutations concerning cyclotomic fields

- Mathematics
- 2019

Let $p$ be an odd prime. For any $p$-adic integer $a$ we let $a\ (p)$ denote the unique integer $x$ with $-p/2 2$ be a positive integers. $\mathbb{Q}(\zeta_n)$ denotes the $n$-th cyclotomic field,…

A conjecture of Zhi-Wei Sun on determinants over finite fields

- Mathematics
- 2021

In this paper, we study certain determinants over finite fields. Let $\mathbb{F}_q$ be the finite field of $q$ elements and let $a_1,a_2,\cdots,a_{q-1}$ be all nonzero elements of $\mathbb{F}_q$. Let…

## References

SHOWING 1-10 OF 32 REFERENCES

Extending the Zolotarev–Frobenius approach to quadratic reciprocity

- Mathematics
- 2015

In 1872, Zolotarev observed that the Legendre symbol $$\left( \frac{a}{p}\right) $$ap is the sign of the permutation of $$\mathbb Z/p\mathbb Z$$Z/pZ induced by multiplication by $$a$$a and used this…

A remark on Zoloterav's theorem

- Mathematics
- 2006

Let n>=3 be an odd integer. For any integer a prime to n, define the permutation gamma_{a,n} of {1,...,(n-1)/2} by gamma_{a,n}(x)=n-\dec{ax}_n if {ax}_n>=(n+1)/2, and {ax}_n if {ax}_n<=(n-1)/2, where…

A Remark on Zoloterav's Theorem

- Mathematics
- 2006

Let n ≥ 3 be an odd integer. For any integer a prime to n, define the permutation γ a,n of {1,. .. , (n − 1)/2} by γ a,n (x) = n − {ax} n if {ax} n ≥ (n + 1)/2, {ax} n if {ax} n ≤ (n − 1)/2, where…

On R. Chapman's "evil determinant": case p=1 (mod 4)

- Mathematics
- 2011

For p=1 (mod 4), we prove the formula (conjectured by R. Chapman) for the determinant of the matrix C with C(i,j)=LegendreSymbol(j-i,p), i,j=0,...,(p-1)/2.

Advanced Number Theory

- Mathematics
- 1980

When a person thinks of algebra, they typically think of a process used to solve polynomial equations. Modern Number theory has evolved through several stages in the past two millennia. Notions of…

Quadratic Reciprocity in a Finite Group

- MathematicsAm. Math. Mon.
- 2005

A key role in our story is played by group characters. Recall that a character X of a finite Abelian group G is a homomorphism from G into C*, the multiplicative group of nonzero complex numbers. The…

A classical introduction to modern number theory

- MathematicsGraduate texts in mathematics
- 1982

This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curve.

Determinants of Legendre symbol matrices

- Mathematics
- 2004

We study and evaluate determinants of various matrices built up from the Legendre symbol defined modulo a prime p. MSC class: 11C20