Editing Symmetric polynomial (section)

==Examples==
The following polynomials in two variables ''X''<sub>1</sub> and ''X''<sub>2</sub> are symmetric:
:<math>X_1^3+ X_2^3-7</math>
:<math>4 X_1^2X_2^2 +X_1^3X_2 + X_1X_2^3 +(X_1+X_2)^4</math>
as is the following polynomial in three variables ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>:
:<math>X_1 X_2 X_3 - 2 X_1 X_2 - 2 X_1 X_3 - 2 X_2 X_3</math>
There are many ways to make specific symmetric polynomials in any number of variables (see the various types below). An example of a somewhat different flavor is
:<math>\prod_{1\leq i<j\leq n}(X_i-X_j)^2</math>
where first a polynomial is constructed that changes sign under every exchange of variables, and taking the [[square (algebra)|square]] renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its [[discriminant]]).

On the other hand, the polynomial in two variables
:<math>X_1 - X_2</math>
is not symmetric, since if one exchanges <math>X_1</math> and <math>X_2</math> one gets a different polynomial, <math>X_2 - X_1</math>. Similarly in three variables
:<math>X_1^4X_2^2X_3 + X_1X_2^4X_3^2 + X_1^2X_2X_3^4</math>
has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial.  However, the following is symmetric:
:<math>X_1^4X_2^2X_3 + X_1X_2^4X_3^2 + X_1^2X_2X_3^4 +
        X_1^4X_2X_3^2 + X_1X_2^2X_3^4 + X_1^2X_2^4X_3</math>