Associated Topics || Dr. Math Home || Search Dr. Math

Symmetric Polynomials

```
Date: 07/31/97 at 05:43:03
From: Jillian Stewart
Subject: Symmetric polynomials

Hello!

I have a question about polynomials and I hope you can help me (no one
else could).

How can it be proved that any symmetric polynomial can be expressed as
an elementary symmetric polynomial?  (I don't even know what symmetric
polynomials are.)

Thank you!
```

```
Date: 07/31/97 at 11:36:48
From: Doctor Wilkinson
Subject: Re: Symmetric polynomials

A symmetric polynomial is one that doesn't change when you permute its
variables. For example, x^2 + y^2 is a symmetric polynomial in x and
y, because if you exchange x and y, the polynomial doesn't change; but
x^2 + 2y^2 isn't symmetric, because if you exchange x and y you get
2x^2 + y^2, which is different.

The elementary symmetric polynomials in x1, x2, x3, x4, ..., xn are

s1 = x1 + x2 + x3 + ... + xn
s2 = x1x2 + x1x3 +... +x2x3 + x2x4 +... + ... +... +xn-1xn
...
sn = x1x2x3...xn

That is, the kth symmetric polynomial is formed by taking all the
possible products of k variables and adding them together.

The theorem you cite is non-trivial but it is not really hard either.
It can be proved by induction on the number of variables and you can
find a proof in a lot of advanced algebra books.  An example of what
it is saying is that

x^2 + y^2 = (x + y)^2 - 2xy = (s1(x, y))^2 - 2s2(x, y)

If you try some other examples, you can sort of figure out how a
general proof has to go, but you have to be pretty good at induction
proofs to get it all to fit together.

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/

```
Associated Topics:
High School Polynomials

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search