Cyclic Subgroups of Multicomplex n-th Roots of Unity

We explore the group structure of n-th roots of unity in multicomplex algebras, focusing on complex (C_1), bicomplex (C_2), and tricomplex (C_3) systems. In the classical complex case, n-th roots of unity form a cyclic group, with each subgroup corresponding to a divisor of n. Extending this to bicomplex and tricomplex algebras, the idempotent decomposition reveals a higher-dimensional structure, resulting in a rapid increase in the number of distinct cyclic subgroups. Explicit formulas, derived via prime-power factorization, and illustrative examples for small n, demonstrate this growth and highlight the combinatorial richness of multicomplex spaces. These results generalize classical group-theoretic properties and provide a framework for analyzing the algebraic and geometric symmetries of higher-dimensional multicomplex systems.