On Algebraic Solutions of Polynomial Equations of Degree n in one Variable

We will show that the roots of a polynomial equation in one variable of degree n are related to the solutions of a symmetric quadratic form in n-1 variables with constant positive integer coefficients. The classic polynomial notation will be rewritten to define a characteristic discriminant of a polynomial of degree n. A new set of characteristic roots allows expressing the characteristic discriminant as the result of a symmetric quadratic form.