Abstract
Recall that D is a half-factorial domain (HFD) when D is atomic and every equation \(\pi _1\cdots \pi _k = \rho _1 \cdots \rho _\ell \), with all \(\pi _i\) and \(\rho _j\) irreducible in D, implies \(k=\ell \). We explain how techniques introduced to attack Artin’s primitive root conjecture can be applied to understand half-factoriality of orders in real quadratic number fields. In particular, we prove that (a) there are infinitely many real quadratic orders that are half-factorial domains, and (b) under the generalized Riemann hypothesis, \({\mathbb {Q}}(\sqrt{2})\) contains infinitely many HFD orders.
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Notes
Here GRH refers to the Riemann hypothesis for all Dedekind zeta functions.
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Acknowledgements
The author is supported by the National Science Foundation (NSF) under award DMS-2001581. He expresses his thanks to Komal Agrawal for helpful discussions around Chen’s paper [3], as well as to the referee for valuable feedback.
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In memory of Kevin James.
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Pollack, P. Half-factorial real quadratic orders. Arch. Math. 122, 491–500 (2024). https://doi.org/10.1007/s00013-024-01969-z
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DOI: https://doi.org/10.1007/s00013-024-01969-z