Editing Ring of symmetric functions (section)

=== A principle relating symmetric polynomials and symmetric functions ===

For any symmetric function ''P'', the corresponding symmetric polynomials in ''n'' indeterminates for any natural number ''n'' may be designated by ''P''(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>). The second definition of the ring of symmetric functions implies the following fundamental principle:

:If ''P'' and ''Q'' are symmetric functions of degree ''d'', then one has the identity <math>P=Q</math> of symmetric functions [[if and only if]] one has the identity ''P''(''X''<sub>1</sub>,...,''X''<sub>''d''</sub>)&nbsp;=&nbsp;''Q''(''X''<sub>1</sub>,...,''X''<sub>''d''</sub>) of symmetric polynomials in ''d'' indeterminates. In this case one has in fact ''P''(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>)&nbsp;=&nbsp;''Q''(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>) for ''any'' number ''n'' of indeterminates.

This is because one can always reduce the number of variables by substituting zero for some variables, and one can increase the number of variables by applying the homomorphisms ''φ''<sub>''n''</sub>; the definition of those homomorphisms assures that ''φ''<sub>''n''</sub>(''P''(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>))&nbsp;=&nbsp;''P''(''X''<sub>1</sub>,...,''X''<sub>''n''+1</sub>) (and similarly for ''Q'') whenever ''n''&nbsp;≥&nbsp;''d''. See [[Newton's identities#Derivation of the identities|a proof of Newton's identities]] for an effective application of this principle.