Monogenic pure cubics

TL;DR

This paper investigates the frequency of monogenic pure cubic fields generated by certain square-free integers and polynomials, providing bounds on their count and assuming ABC conjecture for sharper estimates.

Contribution

It establishes asymptotic bounds for the number of monogenic pure cubic fields over both number fields and finite fields, including unconditional and conditional results.

Findings

Number of such fields grows at least as fast as N^{1/3}

Upper bounds are N divided by a logarithmic factor, improved under ABC

Analogous finite field results are also proven

Abstract

Let $k\geq 2$ be a square-free integer. We prove that the number of square-free integers $m\in [1,N]$ such that $(k,m)=1$ and $\mathbb{Q}(\sqrt[3]{k^2m})$ is monogenic is $\gg N^{1/3}$ and $\ll N/(\log N)^{1/3-\epsilon}$ for any $\epsilon>0$. Assuming ABC, the upper bound can be improved to $O(N^{(1/3)+\epsilon})$. Let $F$ be the finite field of order $q$ with $(q,3)=1$ and let $g(t)\in F[t]$ be non-constant square-free. We prove unconditionally the analogous result that the number of square-free $h(t)\in F[t]$ such that $\deg(h)\leq N$, $(g,h)=1$ and $F(t,\sqrt[3]{g^2h})$ is monogenic is $\gg q^{N/3}$ and $\ll N^2q^{N/3}$.