In mathematics , a polynomial is called symmetrical in several indeterminates , if the indeterminates can be interchanged without changing the polynomial.
on the polynomial ring . A polynomial is symmetric if and only if it is invariant under this operation, i.e. i.e. if
for all
applies. One possible notation for the ring of symmetrical polynomials is therefore
Body of symmetrical functions
We are now replacing the base ring with a base body . The body of symmetric functions is analogous to the above definition of the fixed field below , thus: .
The expansion of the body is Galois with the Galois group and thus has degrees
Examples
The polynomial is symmetric in and , but not symmetric in .
A symmetrical polynomial can be formed from any arbitrary polynomial in the variables by adding the images under the permutations, i.e.:
Elementary symmetric polynomials
A particularly important type of symmetric polynomials are the so-called elementary symmetric polynomials. They are the basic building blocks of symmetrical polynomials in the sense that the latter can always be expressed as a polynomial in the former, and this in only one way.
For every number ( degree of symmetry) of indeterminates and every (polynomial) degree there is exactly one elementary symmetric polynomial .
The two elementary symmetric polynomials in the variables , are
such as
In three variables , , one has the three elementary symmetric polynomials
Power sums
With the power sums
,
for one has another kind of symmetric polynomial. They are connected to the elementary symmetric polynomials via the Newton identities . For example, you have:
And vice versa:
If the ring contains the rational numbers , a similar theorem applies as for the elementary symmetric polynomials:
Every symmetric polynomial can be written as a polynomial in power sums.