The Density of a family of monogenic number fields

TL;DR

This paper investigates the density of primes for which certain monogenic number fields generated by polynomials of the form t^q - p are monogenic, providing lower bounds using the Chebotarev density theorem.

Contribution

It establishes lower bounds on the density of primes p such that t^q - p yields a monogenic number field, including specific results for q=3.

Findings

Density of primes p with monogenic t^q - p is at least (q-1)/q.

For q=3, at least 1/9 of primes p produce non-monogenic fields.

Uses Chebotarev density theorem to derive these density bounds.

Abstract

A monogenic polynomial $f$ is a monic irreducible polynomial with integer coefficients which produces a monogenic number field. For a given prime $q$, using the Chebotarev density theorem, we will show the density of primes $p$, such that $t^q-p$ is monogenic, is bigger or equal than $(q-1)/q$. We will also prove that, when $q=3$, the density of primes $p$, which $\mathbb{Q}(\sqrt[3]{p})$ is non-monogenic, is at least 1/9.