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Monogenic pure cubics. (English) Zbl 1460.11126

J. Number Theory 219, 356-367 (2021); corrigendum ibid. 242, 244 (2023).
Summary: 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 \(\mathbb{Q} (\sqrt[3]{k^2 m})\) 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 \(F(t, \sqrt[3]{g^2 h})\) is monogenic is \(\gg q^{N / 3}\) and \(\ll N^2 q^{N / 3}\).

MSC:

11R16 Cubic and quartic extensions
11R58 Arithmetic theory of algebraic function fields
11D25 Cubic and quartic Diophantine equations
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References:

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