Express power sums in terms of elementary symmetric function

[Yiming Xu]

The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials.

I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do with $k=1,2,3,4$ and an arbitary $n$? Here seems to be a solution, with the usage of a remark of proving the theorem using the algorithm. But I cannot understand what does the remark actually meaning and where does it come from. Could someone explain? Thanks so much!

http://www-users.math.umn.edu/~Garrett/m/algebra/notes/15.pdf

 

[Yiming Xu]

If someone cannot see the notations clearly, it is the same question on SE:http://math.stackexchange.com/quest...ms-in-terms-of-elementary-symmetric-functions