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arXiv · 2111.04215

On the number of monogenizations of a quartic order

Manjul Bhargava
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Abstract

We show that an order in a quartic field has fewer than $3000$ essentially different generators as a $\mathbb Z$-algebra (and fewer than $200$ if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of $2^{72}$. Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most $10$ classes of integral binary quartic forms (and at most $7$ if the discriminant is sufficiently large). This significantly improves the previously best known bound of $2^{80}$.

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Manjul Bhargava. 2021-11-08. On the number of monogenizations of a quartic order. https://arxiv.org/abs/2111.04215

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