THE STRUCTURE OF MULTIPLICATIVE GROUPS OF RESIDUE CLASS RINGS OF THE EISENSTEIN INTEGERS
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Abstract
This paper presents a comprehensive analysis of residue class ring of the Eisenstein integer ring modulo powers of prime elements. As a consequence, we also characterize the structure of the multiplicative group modulo arbitrary (composite) Eisenstein integers. Additionally, we determine all Eisenstein integers that admit primitive roots.
Keywords
Eisenstein integers, multiplicative groups, primitive roots
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References
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Bolker, E. D. (1970). Elementary Number Theory. W. A. Benjamin.
Cross, J. T. (1983). The Euler φ-function in the Gauss integers. American Mathematical Monthly, 90(8), 518-528.
Titu, A., Andreescu, T., & Mushkarov, O. (2017). Number Theory: Concepts and Problems. XYZ Press.
Bolker, E. D. (1970). Elementary Number Theory. W. A. Benjamin.
Cross, J. T. (1983). The Euler φ-function in the Gauss integers. American Mathematical Monthly, 90(8), 518-528.
Titu, A., Andreescu, T., & Mushkarov, O. (2017). Number Theory: Concepts and Problems. XYZ Press.