Monogenic pure cubics

Abstract

Let k≥2 be a square-free integer. We prove that the number of square-free integers m∈[1,N] such that (k,m)=1 and Q(k2m3) is monogenic is ≫N1/3 and ≪N/(log⁡N)1/3−ϵ for any ϵ>0. Assuming ABC, the upper bound can be improved to O(N(1/3)+ϵ). Let F be the finite field of order q with (q,3)=1 and let g(t)∈F[t] be non-constant square-free. We prove unconditionally the analogous result that the number of square-free h(t)∈F[t] such that deg⁡(h)≤N, (g,h)=1 and F(t,g2h3) is monogenic is ≫qN/3 and ≪N2qN/3.