Let be a commutative ring. The fundamental theorem of symmetric polynomials says that any symmetric polynomial in can be expressed uniquely as a polynomial in and .

Recently I was thinking about this along the following lines. Let denote the set of all symmetric polynomials in . Then the theorem above is saying that is generated by as an –algebra, i.e., . Being unsuccessful at utilising this I ended up with the following. (I can’t see the best way of showing that a set generates an algebra.)

Let . Since , we have for all . Hence . So it suffices to show that can be expressed as a polynomial in and for each . But this follows easily by induction and the following identity

.

However, this argument doesn’t generalise immediately to more variables, and I don’t particularly like any of the proofs that I’ve found so far.