power sums are enough for rationality? [closed]

If I have k algebraic integers like a_1, ..., a_k such that the sum of their n-power are integer for n=1, ...m can we deduce that a_1, ..., a_k are integers? how large m should be? (how many power sum should be integers to deduce all a_i's are integers)

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closed as too localized by Gjergji Zaimi, Andreas Blass, Andrés Caicedo, Qiaochu Yuan, Dan PetersenOct 25 '12 at 6:27

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Consider roots of unity. Gerhard "Ask Me About System Design" Paseman, 2012.10.24 – Gerhard Paseman Oct 25 '12 at 3:08
Look at Lucas numbers for example. $$(\frac{1-\sqrt{5}}{2})^n+(\frac{1+\sqrt{5}}{2})^n$$ is always an integer. – Gjergji Zaimi Oct 25 '12 at 3:09