Editing Symmetric polynomial (section)

=== Elementary symmetric polynomials ===
{{Main|Elementary symmetric polynomial}}
For each nonnegative [[integer]] ''k'', the elementary symmetric polynomial ''e''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is the sum of all distinct products of ''k'' distinct variables.  (Some authors denote it by σ<sub>''k''</sub> instead.) For ''k''&nbsp;=&nbsp;0 there is only the [[empty product]] so ''e''<sub>0</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)&nbsp;=&nbsp;1, while for ''k''&nbsp;&gt;&nbsp;''n'', no products at all can be formed, so ''e''<sub>''k''</sub>(''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub>)&nbsp;=&nbsp;0 in these cases. The remaining ''n'' elementary symmetric polynomials are building blocks for all symmetric polynomials in these variables: as mentioned above, any symmetric polynomial in the variables considered can be obtained from these elementary symmetric polynomials using multiplications and additions only. In fact one has the following more detailed facts:
*any symmetric polynomial ''P'' in ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> can be written as a [[polynomial expression]] in the polynomials ''e''<sub>''k''</sub>(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) with 1&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''n'';
*this expression is unique up to equivalence of polynomial expressions;
*if ''P'' has [[integer|integral]] coefficients, then the polynomial expression also has integral coefficients.
For example, for ''n'' = 2, the relevant elementary symmetric polynomials are ''e''<sub>1</sub>(''X''<sub>1</sub>, ''X''<sub>2</sub>) = ''X''<sub>1</sub> + ''X''<sub>2</sub>, and ''e''<sub>2</sub>(''X''<sub>1</sub>, ''X''<sub>2</sub>) = ''X''<sub>1</sub>''X''<sub>2</sub>. The first polynomial in the list of examples above can then be written as
:<math>X_1^3+X_2^3-7=e_1(X_1,X_2)^3-3e_2(X_1,X_2)e_1(X_1,X_2)-7</math>
(for a [[mathematical proof|proof]] that this is always possible see the [[fundamental theorem of symmetric polynomials]]).