Some permutations over ${\mathbb F}_p$ concerning primitive roots

TL;DR

This paper studies permutations induced by primitive roots over finite fields, focusing on their sign and its relation to class numbers of quadratic fields, revealing connections between number theory and permutation properties.

Contribution

It introduces a new permutation based on primitive roots and characterizes its sign in terms of class numbers of quadratic fields, linking permutation properties to algebraic number theory.

Findings

The sign of the permutation relates to the class number of quadratic fields.

Explicit formulas for the permutation's sign when p ≡ 5 mod 8.

Connections established between primitive roots, permutation signs, and class numbers.

Abstract

Let $p$ be an odd prime and let ${\mathbb F}_p$ denote the finite field with $p$ elements. Suppose that $g$ is a primitive root of ${\mathbb F}_p$. Define the permutation $\tau_g:\,{\mathcal H}_p\to{\mathcal H}_p$ by $$ \tau_g(b):=\begin{cases} g^b,&\text{if }g^b\in{\mathcal H}_p,\\ -g^b,&\text{if }g^b\not\in{\mathcal H}_p,\\ \end{cases} $$ for each $b\in{\mathcal H}_p$, where ${\mathcal H}_p=\{1,2,\ldots,(p-1)/2\}$ is viewed as a subset of ${\mathbb F}_p$. In this paper, we investigate the sign of $\tau_g$. For example, if $p\equiv 5\pmod{8}$, then $$ (-1)^{|\tau_g|}=(-1)^{\frac{1}{4}(h(-4p)+2)} $$ for every primitive root $g$, where $h(-4p)$ is the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-4p})$.