A note on prime index of a certain subgroup in $\mathbb{F}_p^*$

TL;DR

This paper investigates the distribution of primes for which a specific subgroup in the multiplicative group of a finite field has an odd prime index, under the assumption of the generalized Riemann hypothesis, revealing a near 46% ratio.

Contribution

It provides an analysis of prime indices of a particular subgroup in finite fields under GRH, highlighting the proportion of primes with odd prime index.

Findings

Approximately 46% of primes have the subgroup with an odd prime index.

The ratio approaches 46% under the generalized Riemann hypothesis.

The study offers insights into subgroup structures in finite fields.

Abstract

Under the generalized Riemann hypothesis, we illustrate that the ratio of the set of primes $p$ such that $\langle -1, 2 \rangle$ has an odd prime index in $\mathbb{F}_p^*$ to the set of primes $p$ such that the subgroup has index greater than $2$ nears $46 \%$.