Symmetric polynomial

In mathematics, a polynomial in several indeterminates is called symmetric if you can swap the indeterminates with each other without changing the polynomial.

For example, considering the polynomials and we obtain by interchanging and the polynomials

Since addition is commutative, so it is replaced in the case of the same polynomial, i.e., is symmetrical and, in the case of a different polynomial is obtained, is not symmetrical.

• 3.1 Definition
• 3.2 Examples
• 3.3 Features
• 3.4 Examples

Formal definition

Let a natural number, a ring. Then a polynomial is called symmetric if

Applies.

Equivalent descriptions are:

• For all

• It should be

• The symmetric group operates through

Body of symmetric functions

The body of the symmetric functions is analogous to the above definition under the fixed field, ie: . The field extension is Galois with Galois group and thus has degree

Examples

• The polynomial is symmetric in and, but not symmetrical.
• For any polynomial in the variables is a symmetric polynomial can be formed by adding the images under the permutations, ie:

Elementary symmetric polynomials

Definition

There were indeterminate. The coefficients of

As a polynomial in are symmetrical; they are called elementary symmetric polynomials. They are explicitly be specified as

Use can also be written as

Examples

• The two elementary symmetric polynomials in the variables,

• In the three variables, there are three elementary symmetric polynomials

Properties

• Fundamental theorem of elementary symmetric polynomials:

• The elementary symmetric polynomials are algebraically independent.

• Let be an integral domain,

Examples

• Generally the power sums with the elementary symmetric polynomials by Newton's identities are connected.
• The polynomial