The proportion of monogenic orders of prime power indices of the pure cubic field

TL;DR

This paper studies the frequency of monogenic orders with prime power indices in pure cubic fields, showing that such orders are rare for primes other than 2 and 3, using algebraic and Diophantine methods.

Contribution

It establishes that the proportion of monogenic orders with prime power indices is zero for primes not equal to 2 or 3, combining order counting and Thue-Mahler equations.

Findings

Proportion of monogenic orders is zero for primes ≠ 2, 3.

Counted orders with prime power indices via submodule analysis.

Bound the number of monogenic orders using solutions to Thue-Mahler equations.

Abstract

In this paper, we investigate the proportion of monogenic orders among the orders whose indices are a power of a fixed prime in a pure cubic field. We prove that the proportion is zero for a prime number that is not equal to 2 or 3. To do this, we first count the number of orders whose indices are power of a fixed prime. This is done by considering every full rank submodules of the ring of integers, and establishing the condition to be closed under multiplication. Next, we derive the index form of arbitrary orders based on the index form of the ring of integers. Then, we obtain an upper bound of the number of monogenic orders with prime power indices by applying the finiteness of the number of primitive solutions of the Thue-Mahler equation.