User:Saung Tadashi/Eilenberg–Niven theorem

Eilenberg–Niven theorem is a theorem of Samuel Eilenberg and Ivan M. Niven. It generalizes the fundamental theorem of algebra to quaternionic polynomials, that is, polynomials with quaternion coefficients and variables.

Statement
Let

$$P(x) = a_0 x a_1 x \cdots x a_n + \phi(x)$$

where x, a1, ..., an are non-zero quaternions and φ(x) is a finite sum of monomials similar to the first term but with degree less than n. Then P(x) has at least one solution.