The Tricomplex Polynomial and Its Root Structure

This paper develops a systematic theory of Tricomplex Polynomials and their roots. It is proved that a polynomial of degree 𝑛 in ℂ3 possesses 𝑛4 roots, generalizing the classical fundamental theorem of algebra. The study further identifies the conditions under which the existence of one root ensures that its conjugate elements are also roots. The quadratic equation 𝜁2 = 𝜂 is solved for various 𝜂 ∈ ℂ3, providing a complete classification of tricomplex square roots. Furthermore, the interrelations among roots, their conjugates, and their norms are analyzed, revealing deep structural analogies with both complex and bicomplex number systems.