Understanding the Fundamental Theorem of Symmetric Polynomials within the context of proving $\pi$ transcendental 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.
 A: Briefly (and perhaps somewhat obviously), symmetric polynomials are useful because they are exactly those which are invariant under all permutations of the variables. There are a lot of situations where we consider what happens if we switch around the roles of the $x_i$. 
The Fundamental Theorem tells us that there is a convenient basis for symmetric polynomials, namely, the "elementary symmetric polynomials" (these are the ones $x_1 + \dotsb + x_n,\, \sum_{i<j}x_ix_j,\, $etc.) 

Example: Suppose $p(z) = a_0 + a_1z + \dotsb + z^n\in \mathbb{F}[x]$ is a polynomial over the complex numbers (or, if you want more generality, over an algebraically closed field). We can write $p(z) = \prod_{i}(z-\alpha_i)$ for $\alpha_i$ the roots. Then I claim the coefficients $a_i$ are the elementary symmetric polynomials in the roots $\alpha_j$. For instance, $a_0 = \prod_i \alpha_ i$ and $a_{n-1} = \alpha _1 + \dotsb + \alpha_n$. Now, the fundamental theorem tells us that if we have any symmetric polynomial in the $\alpha_i$ (for example, the discriminant), then this can be written in terms of the coefficients $a_i$.

Is it the abstract algebra (rings, isomorphisms, etc.) which is confusing you?
Sorry my answer is not more in depth, hope this helps.
A: There is an article entitled "The Powers of Pi are Irrational."  It gives a development of the transcendence of pi using easier irrationality proofs.  Just google the title and it comes up.
