In mathematics, a symmetric polynomial is a polynomial in n variables P(X₁,X₂,...,X_n), such that if some of the variables are interchanged, the polynomial stays the same. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ...

Examples

• 
• P(X₁,X₂) = 4X₁X₂
• P(X₁,X₂,X₃) = X₁X₂X₃ − 2X₁X₂ − 2X₁X₃ − 2X₂X₃

are all symmetric. The polynomial P(X₁,X₂) = X₁ − X₂ is not symmetric, since if we exchange X₁ and X₂ we get the polynomial X₂ − X₁ which is not the same thing.

The building blocks for symmetric polynomials

For each natural number n, there are n so-called elementary symmetric polynomials in the variables X₁, X₂, …, X_n. They are the building blocks for all symmetric polynomials in these variables, meaning that any symmetric polynomial in n variables can be obtained from the elementary symmetric polynomials via several multiplications and additions. More precisely: any symmetric polynomial in n variables is a polynomial of the n+1 elementary symmetric polynomials in these variables. For example, for n = 2, there are only 3 elementary symmetric polynomials: the constant polynomial 1, the polynomial X₁ + X₂, and the polynomial X₁X₂. The first polynomial in the list of examples above can then be written as In mathematics, elementary symmetric polynomials are basic building block for symmetric polynomials. ...

P(X₁,X₂) = (X₁ + X₂)³ − 3X₁X₂(X₁ + X₂) − 7.

Symmetric polynomials in algebra

Symmetric polynomials are important to linear algebra, representation theory, and Galois theory. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In mathematics, Galois theory is a branch of abstract algebra. ...

See also

• symmetric function — this term often refers to a symmetric polynomial
• Newton's identities

In mathematics, the theory of symmetric functions is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. ... In mathematics, Newtons identities relate two different ways of describing the roots of a polynomial. ...

References

• S. Lang, Algebra, Springer-Verlag, 2004, ISBN 0-387-95385-X.