power sums are enough for rationality? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T13:38:23Z http://mathoverflow.net/feeds/question/110611 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110611/power-sums-are-enough-for-rationality power sums are enough for rationality? katie 2012-10-25T03:04:57Z 2012-10-25T04:52:40Z <p>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)</p> http://mathoverflow.net/questions/110611/power-sums-are-enough-for-rationality/110617#110617 Answer by Aaron Meyerowitz for power sums are enough for rationality? Aaron Meyerowitz 2012-10-25T04:41:20Z 2012-10-25T04:41:20Z <p>Take any monic polynomial with integer coefficients and look at the roots.</p> http://mathoverflow.net/questions/110611/power-sums-are-enough-for-rationality/110618#110618 Answer by Geoff Robinson for power sums are enough for rationality? Geoff Robinson 2012-10-25T04:52:40Z 2012-10-25T04:52:40Z <p>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.</p>