On a simple derivation of the noncommutative vieta theorem

Inspired by the works of Gelfand and Retakh, we present here a simple derivation of the noncommutative Vieta theorem. It is not necessary to invoke the theory of quasideterminants and noncommutative symmetric functions for our proof. We just use the method of induction. We have also demonstrated that the formulas for the coefficients are independent of the ordering of the roots.