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

Abstract

Given a real parameter 𝛼 and a semigroup 𝑆 , we consider a functional equation arising from the multiplicative structure of the quartic number field 𝑄 ( 𝛼 4 ) . 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.