Theory of equations/Symmetric functions of roots

Symmetric functions of roots arise naturally in the Theory of equations and provide a powerful way to relate the solutions of a polynomial to its coefficients. For a polynomial with given roots, any expression in the roots that remains unchanged under all permutations of those roots is called a symmetric function. These functions lie at the heart of classical elimination theory and modern algebra, including Galois theory and the theory of symmetric polynomials.G. Chrystal, Algebra, Part II: Theory of Equations, Adam and Charles Black (1900), ch. VI – Symmetric functions of the roots.C. Jordan, Traité des substitutions et des équations algébriques, Gauthier-Villars (1870), sections on symmetric functions.B. L. van der Waerden, Modern Algebra, Springer (1930), chap. VI. Basic setup Consider a polynomial of degree with (not necessarily distinct) roots in some extension field of the coefficient field. Then A function is called a symmetric function of the roots if its value is unchanged when the roots are permuted: for every permutation in the symmetric group .M. Hazewinkel (ed.), Handbook of Algebra, Vol. 3, North-Holland (2003), article Symmetric functions. Important examples include: the sum of the roots ; the sum of pairwise products ; the product ; more complicated expressions such as or , which are symmetric but not elementary. Elementary symmetric functions The most fundamental symmetric functions are the elementary symmetric functions in the roots, defined by Thus If we expand , comparison of coefficients with the standard form gives the classical relations: These formulas are often referred to as Viète's relations or Viète's formulae, after François Viète, who first systematically expressed polynomial coefficients through symmetric functions of the roots.J. Stillwell, Mathematics and Its History, 3rd ed., Springer (2010), pp. 150–153. General symmetric functions and the fundamental theorem Any polynomial expression in the roots that is symmetric can be written as a polynomial in the elementary symmetric functions. This is the content of the fundamental theorem on symmetric polynomials: Every symmetric polynomial with coefficients in a commutative ring can be expressed uniquely as a polynomial in the elementary symmetric polynomials with coefficients in . As a consequence, every symmetric function of the roots of a polynomial with coefficients in can ultimately be written in terms of the coefficients of the polynomial itself.I. N. Herstein, Topics in Algebra, 2nd ed., Wiley (1975), chap. 5.M. Artin, Algebra, 2nd ed., Pearson (2011), sec. 12.2. For example, if are roots of , then is symmetric, and one can compute Expressing by the coefficients of gives a formula for the sum of squares of the roots in terms of those coefficients. Power sums and Newton's identities Another important family of symmetric functions consists of the power sums The power sums are also symmetric polynomials, and they are related to the elementary symmetric functions by a system of recursive relations called Newton's identities (or Newton–Girard formulas). For they can be written in the form where for .M. Marcus, Finite Dimensional Multilinear Algebra, Part I, Marcel Dekker (1973), sec. 5.3. In practice, these identities allow one to compute sums such as from the coefficients of the polynomial, by first expressing the from Viète's relations and then finding the recursively. Symmetric functions and relations between roots Symmetric functions of the roots are particularly useful for: deriving constraints on the roots when the coefficients are known; constructing new equations satisfied by expressions in the roots; studying how the roots change under algebraic operations such as scaling or translation. For instance, if and are roots of a quadratic polynomial then and hence This expresses a more complicated symmetric function directly in terms of the coefficients , without solving the equation explicitly.R. Barnard and J. Child, A Short Account of the History of Symmetric Functions of the Roots of an Algebraic Equation, The Mathematical Gazette, Vol. 9, No. 134 (1918), pp. 137–149.Vedantu, Theory of Equations – Symmetric Functions of Roots, JEE Main preparation notes, accessed 2025-12-22. Applications Symmetric functions of roots appear in many areas of mathematics and its applications: In algebraic equation theory, they form the basis of classical solution methods and elimination procedures. In Galois group computations, invariants built from symmetric functions help determine field extensions and solvability by radicals. In invariant theory and the representation theory of the symmetric group, symmetric functions such as the Schur functions play a central role. In combinatorics, generating functions built from symmetric polynomials encode counting problems related to partitions and Young diagrams. In numerical analysis, relations between symmetric functions and coefficients are used to study the sensitivity of roots to perturbations of coefficients. In undergraduate courses on the theory of equations, problems involving symmetric functions of roots typically ask for expressions such as , or in terms of the coefficients of the given polynomial, often using Viète's relations and Newton's identities as main tools.Department of Mathematics, IIT Kharagpur, Theory of Equations: Symmetric Functions of Roots, course notes (archived PDF). Category:Book:Theory of equations Category:Algebra fr:Theory of equations/Symmetric functions of roots de:Theory of equations/Symmetric functions of roots