"M is a maximal ideal in the polynomial ring k[x_1, x_2, ..., x_n] if and only if M = (x_1 - a_1, ..., x_n - a_n) for some a_1, ..., a_n in k.More specifically, when solving systems of n-variable polynomial equations over the complex field, there exists a solution if the polynomials form a proper ideal in C[x_1, ..., x_n]." . . . .