Noetherian ring of symmetric polynomials

Question

I wish to show that $k[x_1,x_2,..,x_n]^{\Sigma_n}$, which is the ring of all symmetric polynomials, is Noetherian.

I thought the easiest way to do this would be to show that every ideal is finitely generated, but I cant seem to get any proof to hold, any help?

Answer 1

The ring of symmetric polynomials is a polynomial ring generated by elementary symmetric polynomials, hence is Noetherian by Hilbert basis theorem.