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If Z is a sum of t distinct roots of unity and |Z| is a rational integer, can someone find a bound on |Z| in terms of k=deg(Q(Z):Q))?

Clearly we need to have distinct roots of unity otherwise this won't work!

Correction: Let assume that Z is not rational itself otherwise obviously it's wrong. Here I hope to extend the proof of Kronecker thm! I have "Z is a sum of t distinct roots of unity and |Z| is a rational integer" I conjecture that either Z is rational or a root of unity!

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  • $\begingroup$ Not sure what you are asking but you might want to check: Loxton, J. H. On two problems of R. W. Robinson about sums of roots of unity. Acta Arith. 26 (1974/75), 159–174. $\endgroup$ Oct 9, 2012 at 2:04
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    $\begingroup$ Even with the correction, all you have to do is multiply Will Sawin's sum by, say, a nonreal cube root of 1, $\omega$. Then $Z=-n\omega$ is not rational, $|Z|=n$ is a rational integer, and $k=2$. $\endgroup$ Oct 10, 2012 at 4:06
  • $\begingroup$ @Gerry: sorry I edited my answer to fix it before seeing your identical comment. $\endgroup$
    – Will Sawin
    Oct 14, 2012 at 17:20

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If you multiply all the primitive roots of unity for the first $n$ primes by $-i$, then add them, you get $in$, whose degree is $2$ and whose absolute value is $n$, rational and unbounded.

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    $\begingroup$ Dear Will, Doesn't the question ask that $|Z|$ (where $Z$ is the sum of roots of unity in question) be an integer? Regards, $\endgroup$
    – Emerton
    Oct 9, 2012 at 2:25
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    $\begingroup$ I edited it to a more elegant solution that deals with that issue. $\endgroup$
    – Will Sawin
    Oct 9, 2012 at 3:17
  • $\begingroup$ In fact, this construction allows you to write any sum of roots of unity as a sum of distinct roots of unity. $\endgroup$
    – Will Sawin
    Oct 14, 2012 at 22:46

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