Algebraic equations and polynomials over the ring of $p$-complex numbers

In this paper, we study the algebraic equations over the ring of $p$-complex numbers. Remainder division theorems and an analogue of Bezout’s theorem for $p$-complex polynomials are represented. For equations of the 2nd and 3rd degrees, conditions for the existence of roots are obtained, in some cases solutions are given in an explicit form. For polynomials of an arbitrary degree with an invertible leading coefficient, theorems on factorisation with a unit leading coefficient are proven in the cases where there are simple roots, multiple roots, and no roots. It is shown that in the absence of multiple roots, this decomposition will be unique, and in the case of the presence of multiple roots, the polynomial admits an infinite number of expansions.