Monogenic pure cubics

2020/09/05 by Aygin, Zafer Selcuk, Nguyen, Khoa D. · 1 citation
#11R58. Secondary: 11D25 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11R16

paper · doi:10.48550/arxiv.2009.02442

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(√[3]k²m) is monogenic is ≫ N^1/3 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 °(h)≤ N, (g,h)=1 and F(t,√[3]g²h) is monogenic is ≫ q^N/3 and ≪ N²q^N/3.

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