## 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)

-
Consider roots of unity. Gerhard "Ask Me About System Design" Paseman, 2012.10.24 – Gerhard Paseman Oct 25 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 at 3:09