Editing Symmetric polynomial (section)

== Relation with the roots of a monic univariate polynomial ==
Consider a monic polynomial in ''t'' of degree ''n''

:<math>P=t^n+a_{n-1}t^{n-1}+\cdots+a_2t^2+a_1t+a_0</math>

with coefficients ''a''<sub>''i''</sub> in some field&nbsp;''K''. There exist ''n'' roots ''x''<sub>1</sub>,...,''x''<sub>''n''</sub> of ''P'' in some possibly larger field (for instance if ''K'' is the field of [[real number]]s, the roots will exist in the field of [[complex number]]s); some of the roots might be equal, but the fact that one has ''all'' roots is expressed by the relation

:<math>P = t^n+a_{n-1}t^{n-1}+\cdots+a_2t^2+a_1t+a_0=(t-x_1)(t-x_2)\cdots(t-x_n).</math>

By comparing coefficients one finds that
:<math>\begin{align}
a_{n-1}&=-x_1-x_2-\cdots-x_n\\
a_{n-2}&=x_1x_2+x_1x_3+\cdots+x_2x_3+\cdots+x_{n-1}x_n = \textstyle\sum_{1\leq i<j\leq n}x_ix_j\\
& {}\  \, \vdots\\
a_1&=(-1)^{n-1}(x_2x_3\cdots x_n+x_1x_3x_4\cdots x_n+\cdots+x_1x_2\cdots x_{n-2}x_n+x_1x_2\cdots x_{n-1})
      = \textstyle(-1)^{n-1}\sum_{i=1}^n\prod_{j\neq i}x_j\\
a_0&=(-1)^nx_1x_2\cdots x_n.
\end{align}</math>

These are in fact just instances of [[Vieta's formulas]]. They show that all coefficients of the polynomial are given in terms of the roots by a symmetric [[polynomial expression]]: although for a given polynomial ''P'' there may be qualitative differences between the roots (like lying in the base field&nbsp;''K'' or not, being [[simple root|simple]] or multiple roots), none of this affects the way the roots occur in these expressions.

Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing ''P'', and considering them as indeterminates rather than as constants in an appropriate field; the coefficients ''a''<sub>''i''</sub> then become just the particular symmetric polynomials given by the above equations. Those polynomials, without the sign <math>(-1)^{n-i}</math>, are known as the [[elementary symmetric polynomial]]s in ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>. A basic fact, known as the '''[[fundamental theorem of symmetric polynomials]]''', states that ''any'' symmetric polynomial in ''n'' variables can be given by a polynomial expression in terms of these elementary symmetric polynomials. It follows that any symmetric polynomial expression in the roots of a monic polynomial can be expressed as a polynomial in the ''coefficients'' of the polynomial, and in particular that its value lies in the base field ''K'' that contains those coefficients. Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than ''K'' in which those roots may lie. In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are [[Newton's identities]], which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.