Question

Need help, please
Show that the set of symmetric polynomials in 3 variables is a subring of all polynomials in 3 variables.

Answer 1

I need to arrange the form of symmetric polynomial in general. I can handle the proof. :)

Answer 2

@eliassaab

Answer 3

My attempt: let S ={ symmetric polynomials in 3 variables}
P={ polynomials in 3 variables}
then \(S\subset P\) and we know that P is a ring, hence to prove S is a subring, we need prove:
a) S is closed under addition and additive inverse
b) S is closed under multiplication
both a, b are not hard to do, just have problem with generalize the form of symmetric polynomials in 3 variables. :)

Answer 4

Let \(f,g\in S\) then we have 6 forms of f, g they are all equal. They are
f(x,y,z) = f(x, z, y)=f(y, x, z)= f(y, z, x)= f(z, x, y)= f(z, y, x)
g(x,y,z) =g(x, z, y)=g(y, x, z)= g(y, z, x)= g(z, x, y)= g(z, y, x)

Answer 5

Do I have to expand them ( if I do, how?) or just let them as they are and start proving?

Answer 6

damn rings heheh