On the Solution of Algebraic Equations of Small Degrees by Square Radicals

Abstract

The work is devoted to the search for constructive analytical expressions for the roots of algebraic equations of the third-sixth degree by the coefficients of the equations. Relationships are obtained for the coefficients at which the roots of the equations are represented most simply, for example, rationally. Rational expressions for multiple roots are given. A condition is found under which the polynomial of the sixth degree in the canonical form can be represented by the product of polynomials of the third degree in the canonical form. Particular attention is paid to the symbolic expression of the roots of equations by square radicals from the coefficients. A method for solving equations using defining (generating, related to the original) equations is proposed. All the presented expansions are true of polynomials with arbitrary complex coefficients.