<s>[INST] Given $x+y+z=1,x^2+y^2+z^2=2,x^3+y^3+z^3=3,$ can we conclude that $x^n+y^n+z^n\in\mathbb{Q}$ for all $n\in\mathbb{N}$. Is there an explicit form? I had seen a fun problem that is exactly the problem in the question, except it was a specific case of this. Turns out, if $x+y+z=1,x^2+y^2+z^2=2,x^3+y^3+z^3=3,$ then $x^5+y^5+z^5 = 6$. I put this equation into Wolfram Alpha for different values of $n$. For instance, $x^8+y^8+z^8=\frac{51}{72}$. $x^{11}+y^{11}+z^{11}=\frac{11117}{216}.$
Firstly, is there an explicit expression to evaluate $x^n+y^n+z^n$? And, if not, are we able to prove that this is always a rational number, at least? WolframAlpha is unable to calculate an explicit form for $x^n+y^n+z^n$.
 [/INST] Yes.  In https://en.m.wikipedia.org/wiki/Symmetric_polynomial, look at Power-sum Symmetric Polynomials in the section Special kinds of symmetric polynomials
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