I am currently studying the proof of the transcendence of $\pi$. There are a bunch of proofs scattered across the web (here, here, and here, to list some); some derive from the Lindemann-Weierstrass Theorem, while some are given standalone. But all of them take essentially the same approach, perhaps with some rewording.
My question is not on the proof as a whole, but on a particular segment. It would appear that a key element here is a (double) application of the Fundamental Theorem of Symmetric Polynomials. I have tried to understand this concept using info from Wikipedia, but I cannot quite wrap my head around it, or how exactly it works out in the proof.
I was hoping someone could give me a graspable explanation of this theorem, and perhaps an elucidation as to how it applies to the proof. It would also help a lot if someone could give me an application of this theorem to an easy example (i.e., using it outside the context of the proof) so I can get a concrete feel for it. Thanks.