Vieta's formulae for regular polynomials of a quaternionic variable

Given a polynomial p ∈ F[x], with F a commutative ring, classical Vieta’s Formulae explicitely determine the coefficents of p in terms of the roots of p itself. In this paper, Vieta’s For- mulae are obtained for slice–regular polynomials over the non commutative algebra of Quaternions, by applying an argument which essentially relies on the method of induction and without invoking the general theory of quasideterminants and noncommutative symmetric functions.