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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

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    It is not true that $p_k$ will be a sum of $e_k$s - it will be a *polynomial* in the $e_k$s. You say you don't understand the algorithm in the link, but fail to say what part of the algorithm you don't understand. It even gives you a couple examples of using the algorithm. – 2017-02-26
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    @arctictern Thanks a lot, I have understand it. – 2017-02-27

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