Editing Symmetric polynomial (section)

=== Power-sum symmetric polynomials ===
{{Main|Power sum symmetric polynomial}}
For each integer ''k''&nbsp;≥&nbsp;1, the monomial symmetric polynomial ''m''<sub>(''k'',0,...,0)</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is of special interest. It is the power sum symmetric polynomial, defined as
:<math>p_k(X_1,\ldots,X_n) = X_1^k + X_2^k + \cdots + X_n^k .</math> 
All symmetric polynomials can be obtained from the first ''n'' power sum symmetric polynomials by additions and multiplications, possibly involving [[rational number|rational]] coefficients.  More precisely,
:Any symmetric polynomial in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> can be expressed as a polynomial expression with rational coefficients in the power sum symmetric polynomials ''p''<sub>1</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>), ..., ''p''<sub>''n''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>).
In particular, the remaining power sum polynomials ''p''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) for ''k''&nbsp;&gt;&nbsp;''n'' can be so expressed in the first ''n'' power sum polynomials; for example
:<math>p_3(X_1,X_2)=\textstyle\frac32p_2(X_1,X_2)p_1(X_1,X_2)-\frac12p_1(X_1,X_2)^3.</math>

In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in ''n'' variables with ''integral'' coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials.
For an example, for ''n''&nbsp;=&nbsp;2, the symmetric polynomial
:<math>m_{(2,1)}(X_1,X_2) = X_1^2 X_2 + X_1 X_2^2</math>
has the expression
:<math> m_{(2,1)}(X_1,X_2)= \textstyle\frac12p_1(X_1,X_2)^3-\frac12p_2(X_1,X_2)p_1(X_1,X_2).</math>
Using three variables one gets a different expression
:<math>\begin{align}m_{(2,1)}(X_1,X_2,X_3) &= X_1^2 X_2 + X_1 X_2^2 + X_1^2 X_3 + X_1 X_3^2 + X_2^2 X_3 + X_2 X_3^2\\
  &= p_1(X_1,X_2,X_3)p_2(X_1,X_2,X_3)-p_3(X_1,X_2,X_3).
\end{align}</math>
The corresponding expression was valid for two variables as well (it suffices to set ''X''<sub>3</sub> to zero), but since it involves ''p''<sub>3</sub>, it could not be used to illustrate the statement for ''n''&nbsp;=&nbsp;2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first ''n'' power sum polynomials involves rational coefficients may depend on ''n''. But rational coefficients are ''always'' needed to express elementary symmetric polynomials (except the constant ones, and ''e''<sub>1</sub> which coincides with the first power sum) in terms of power sum polynomials. The [[Newton identities]] provide an explicit method to do this; it involves division by integers up to ''n'', which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a [[field (mathematics)|field]] of finite [[characteristic (algebra)|characteristic]]; however, it is valid with coefficients in any [[ring (mathematics)|ring]] containing the rational numbers.