If I have k algebraic integers like a_1, ..., a_k such that the sum of their npower 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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A common theme to all comments and answers prior to this is the fact that if the algebraic integers you start with are closed under algebraic conjugation, then the power sums are all necessarily (by Galois theory) rational, so they are always (rational) integers, as rational algebraic integers are integers. 


Take any monic polynomial with integer coefficients and look at the roots. 

