Korean J. Math. Vol. 33 No. 4 (2025) pp.369-379
DOI: https://doi.org/10.11568/kjm.2025.33.4.369

A Reexamination of Semigroup Homomorphisms Arising from the Product Structure in Pure Quartic Number Fields

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Published: Dec 30, 2025
Keywords:
Functional equations, Homomorphisms, Quartic number field, Multiplicative functions, Semigroups
Mathematics Subject Classification:
39B32, 39B52, 47A62

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Wutiphol Sintunavarat
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathum Thani 12120, Thailand
Pratchaya Singjanusong
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathum Thani 12120, Thailand

Abstract

Given a real parameter $\alpha$ and a semigroup $S$, we consider a functional equation arising from the multiplicative structure of the quartic number field $\mathbb{Q}(\sqrt[4]{\alpha})$. A recent study by Mouzoun and Zeglami [Bol. Soc. Mat. Mex. 28:73 (2022)] investigated solutions to this equation, yet specific results were found to be incorrect. In this work, we provide a rigorous reexamination of the equation, identify the inaccuracies, and introduce refined conditions that ensure its correct formulation. Our results offer a precise characterization of semigroup homomorphisms associated with the product structure in pure quartic number fields, thereby contributing to the broader study of functional equations in algebraic systems.



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