# Symmetric Polynomial Explained for Homework

• storm4438
In summary, a symmetric polynomial is a polynomial where swapping the variables does not change the polynomial. This is useful in understanding Newton's theorem of symmetric polynomials, which is used in Edwards' book on Galois theory. It states that every symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials.
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.

## The Attempt at a Solution

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).

----

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.

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.

## What is a symmetric polynomial?

A symmetric polynomial is a type of algebraic expression that remains unchanged when the variables are permuted or swapped. This means that if we switch the order of the variables, the polynomial will still have the same value.

## How do you determine if a polynomial is symmetric?

To determine if a polynomial is symmetric, we can use the following property: If we replace all occurrences of one variable with another variable, and the polynomial remains unchanged, then it is symmetric.

## What is the significance of symmetric polynomials?

Symmetric polynomials are used in a variety of fields, including algebra, geometry, and combinatorics. They have important applications in areas such as solving polynomial equations, graph theory, and coding theory.

## What are some examples of symmetric polynomials?

Examples of symmetric polynomials include x2 + y2, x3 + y3 + z3, and x4 + 2xy + y4. These polynomials remain unchanged when we swap the variables, for example, if we switch x and y in the first polynomial, we still get x2 + y2.

## How are symmetric polynomials used in mathematics?

Symmetric polynomials are used to study and solve equations with multiple variables, as well as to simplify calculations in various mathematical problems. They also have applications in areas such as group theory, differential equations, and physics.

• Calculus and Beyond Homework Help
Replies
4
Views
234
• Calculus and Beyond Homework Help
Replies
4
Views
3K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
24
Views
2K
• Calculus and Beyond Homework Help
Replies
27
Views
5K
• Calculus and Beyond Homework Help
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
12
Views
2K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
19
Views
3K
• Calculus and Beyond Homework Help
Replies
7
Views
2K