TITLE: Proofs of The Fundamental Theorem of Symmetric Polynomials
QUESTION [12 upvotes]: I have been considering a few proofs of this theorem, and I noticed that a few of them (for example Proof 1, and Proof 2) prove the theorem first for homogeneous symmetric polynomials and then generalise it. However I have seen other proofs which don't seem to require this step (e.g. Proof 3) and it seems to me superfluous.
Could anyone explain why so many proofs include this step and/or if it is necessary?

REPLY [6 votes]: The lex order proof starts with a symmetric polynomial $f_0$. It then subtracts off something to make a new symmetric polynomial $f_1$ whose leading term is less than that of $f_0$. Then we make another symmetric polynomial $f_2$ whose leading term is less than $f_1$, and so on. You need to know that this process stops. In other words, you need to show that lex order has no infinite descending chains.
This isn't so hard to prove for lex order. But it is immediate if you restrict to the homogenous case first, since there are only finitely many monomials of a given degree.