Multiplicative orders of Gauss periods and the arithmetic of real quadratic fields

TL;DR

This paper explores the divisibility properties of multiplicative orders of specific elements in finite fields, connecting these properties to the arithmetic of real quadratic fields to derive new insights.

Contribution

It introduces novel divisibility conditions for multiplicative orders in finite fields by leveraging the relationship with real quadratic fields.

Findings

Derived new divisibility conditions for multiplicative orders

Linked finite field elements to real quadratic field arithmetic

Provided theoretical results connecting algebraic structures

Abstract

We obtain divisibility conditions on the multiplicative orders of elements of the form $\zeta + \zeta^{-1}$ in a finite field by exploiting a link to the arithmetic of real quadratic fields.