Abel's Theorem

Abel's Theorem states that for certain types of equations called algebraic equations, the sum of their solutions (or roots) remains constant when the coefficients of the equation vary along a continuous path, as long as the roots stay well-defined and do not collide or disappear. In essence, if you smoothly change the coefficients of a polynomial, the sum of its roots changes in a predictable way, determined by the coefficients themselves, and this sum remains consistent throughout such transformations. This theorem highlights a deep connection between the roots of an equation and its coefficients.