Let say Z is a sum of nroots of unity and thus an algebraic integer, and D is an rational integer. If z+D is an integer, what can we conclude regarding Z? can we say Z is integer? Another related question is, for which nonzero D, we can conclude Z is integer from the given that Z+D is an integer?
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If $D$ is a nonzero rational number and $R$ is a positive number, the complex numbers $z$ with $z+D=R$ form the circle of radius $R$ centred at $D$. The intersection of this with the circle $z = k$ (if nonempty) consists of one or two points satisfying the quadratic $D z^2 + (D^2 + k^2  R^2) z + k^2 D = 0$. So if $z$ with $z+D^2$ rational has degree $> 2$ over the rationals, $z^2$ can't be a rational number. 

