Some permutations over Fp concerning primitive roots

• Published in: The Ramanujan journal Vol. 55; no. 1; pp. 337 - 348
• Main Authors: Wang, Li-Yuan, Pan, Hao
• Format: Journal Article
• Language: English
• Published: New York Springer US 2021; Springer Nature B.V
• Subjects: Combinatorics; Field Theory and Polynomials; Fields (mathematics); Fourier Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; Number Theory; Permutations; 11R29; Class number; Permutation; 12E20; Secondary 05A05; Primary 11T22; Primitive root of a prime
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-020-00277-8

Let p be an odd prime and let F p denote the finite field with p elements. Suppose that g is a primitive root of F p . Define the permutation τ g : H p → H p by τ g ( b ) : = g b if g b ∈ H p , - g b if g b ∉ H p , for each b ∈ H p , where H p = { 1 , 2 , … , ( p - 1 ) / 2 } is viewed as a subset of F p . In this paper, we investigate the sign of τ g . For example, if p ≡ 5 ( mod 8 ) , then ( - 1 ) | τ g | = ( - 1 ) 1 4 ( h ( - 4 p ) + 2 ) for every primitive root g , where h ( - 4 p ) is the class number of the ring of integers of the imaginary quadratic field Q ( - 4 p ) .