Symmetric Polynomial Explained for Homework

[storm4438]

Homework Statement

No problem exactly I am just reading a book that refrences symmetric polynomials but i don't 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.

Homework Equations

The Attempt at a Solution

 

[gb7nash]

Suppose you have a polynomial in n variables: (X₁, X₂, ..., X_n). 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.

 

[storm4438]

Yes thank you that helps very much, the examples make it much more clear. Now if you don't 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.