When does the modulus of a sum of an integer and an algebraic integer equal an integer? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:18:25Z http://mathoverflow.net/feeds/question/107328 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107328/when-does-the-modulus-of-a-sum-of-an-integer-and-an-algebraic-integer-equal-an-in When does the modulus of a sum of an integer and an algebraic integer equal an integer? katie 2012-09-16T16:11:26Z 2012-12-12T01:38:01Z <p>Let say Z is a sum of n-roots 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 non-zero D, we can conclude |Z| is integer from the given that |Z+D| is an integer?</p> http://mathoverflow.net/questions/107328/when-does-the-modulus-of-a-sum-of-an-integer-and-an-algebraic-integer-equal-an-in/107335#107335 Answer by Robert Israel for When does the modulus of a sum of an integer and an algebraic integer equal an integer? Robert Israel 2012-09-16T19:02:20Z 2012-09-16T19:02:20Z <p>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.</p>