On generalized Legendre matrices involving roots of unity over finite fields

2023/05/25 by Ning-Liu Wei, Yubo Li, Wei, Ning-Liu +3
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2305.16064

openalex publication_date 2023/05/25 · openalex created_date 2023/05/27 · openalex updated_date 2026/07/28

Abstract

In this paper, motivated by the work of Chapman, Vsemirnov and Sun et al., we investigate some arithmetic properties of the generalized Legendre matrices over finite fields. For example, letting a₁,⋯,a_(q-1)/2 be all non-zero squares in the finite field \mathbbF_q which contains q elements with 2\nmid q, we give the explicit value of D_(q-1)/2=det[(a_i+a_j)^(q-3)/2]_{1≤ i,j≤ (q-1)/2}. In particular, if q=p is a prime greater than 3, then (\fracdet D_(p-1)/2p)= \begincases 1 amp; if p≡1\pmod4, (-1)^(h(-p)+1)/2 amp; if p≡ 3\pmod4 and pgt;3, \endcases where (⋅/p) is the Legendre symbol and h(-p) is the class number of ℚ(√(-p)).

Related

• Problems and results on determinants involving Legendre symbols
• Determinants concerning Legendre symbols
• The Pell sequence and cyclotomic matrices involving squares over finite fields
• Gaussian hypergeometric functions and cyclotomic matrices
• Evaluation of a determinant involving Legendre symbols