# Homework Help: Symmetric Polynomial

1. Jun 19, 2011

### storm4438

1. The problem statement, all variables and given/known data

No problem exactly im just reading a book that refrences symmetric polynomials but i dont know what a symmetric polynomial is. I looked at the wiki page but i didn't really get what it was saying. Any help on clearing up the meaning would be greatly appreciated.

2. Relevant equations

3. The attempt at a solution

2. Jun 19, 2011

### gb7nash

Suppose you have a polynomial in n variables: (X1, X2, ..., Xn). e.g. y = X15 + X22 + 2

A symmetric polynomial is a polynomial such that if you swap any of the variables, you still get the same polynomial. For example, consider:

X12 + X2 (1)

Let's swap X1 with X2, so we get:

X22 + X1. This is not equal to (1).

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Now consider:

X12 + X22 + X32 (2)

Let's swap X1 with X2, so we get:

X22 + X12 + X32, which is equivalent with (2). You can further check that no matter how you permute X1, X2 and X3, you will get the same polynomial. Therefore, (2) is a symmetric polynomial.

Hopefully this helped.

3. Jun 19, 2011

### storm4438

Yes thank you that helps very much, the examples make it much more clear. Now if you dont mind me asking one more question. What is Newtons theorem of symmetric polynomials. They use it in Edwards book on galois theory but i didnt understand his explination of it.