2019/08/20 by Hai-Liang Wu, Wu, Hai-Liang
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #Combinatorics #FOS: Mathematics #Mathematics #Modulo #Number Theory (math.NT) #Physics #Primitive root modulo n #math.NT
paper · pdf · doi:10.48550/arxiv.1908.07641
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2019/08/20 · arxiv created 2020/02/03 · arxiv updated 2020/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p=2n+1 be an odd prime, and let ζp2-1 be a primitive (p2-1)-th root of unity in the algebraic closure ℚp of ℚp. We let g∈ℤp[ζp2-1] be a primitive root modulo pℤp[ζp2-1] with g≡ ζp2-1\pmod pℤp[ζp2-1]. Let Δ≡3\pmod4 be an arbitrary quadratic non-residue modulo p in ℤ. By the Local Existence Theorem we know that ℚp(√Δ)=ℚp(ζp2-1). For all x∈ℤ[√Δ] and y∈ℤp[ζp2-1] we use x and y to denote the elements x\mod pℤ[√Δ] and y\mod pℤp[ζp2-1] respectively. If we set ak=k+√Δ for 0≤ k≤ p-1, then we can view the sequence S := a02, ⋯, a02n2, ⋯,ap-12, ⋯, ap-12n2⋯, 12, ⋯,n2 as a permutation σ of the sequence S^* := g2, g4, ⋯,gp2-1. We determine the sign of σ completely in this paper.