Editing Root of unity (section)

==Relation to quadratic integers==
[[Image:Roots of unity, golden ratio.svg|thumb|right|In the [[complex plane]], the red points are the fifth roots of unity, and the black points are the sums of a fifth root of unity and its complex conjugate.]]
[[Image:Star polygon 8-2.svg|thumb|right|160px|In the complex plane, the corners of the two squares are the eighth roots of unity]]<!-- a better image needed -->
For {{math|1=''n'' = 1, 2}}, both roots of unity {{num|1}} and {{num|−1}} are [[integer]]s.

For three values of {{mvar|n}}, the roots of unity are [[quadratic integer]]s:
* For {{math|1=''n'' = 3, 6}} they are [[Eisenstein integer]]s ({{math|1=[[discriminant|''D'']] = −3}}).
* For {{math|1=''n'' = 4}} they are [[Gaussian integer]]s ({{math|1=''D'' = −1}}): see [[Imaginary unit]].

For four other values of {{mvar|n}}, the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its [[complex conjugate]] (also an {{mvar|n}}th root of unity) is a quadratic integer.

For {{math|1=''n'' = 5, 10}}, none of the non-real roots of unity (which satisfy a [[quartic equation]]) is a quadratic integer, but the sum {{math|1=''z'' + ''{{overline|z}}'' = 2 [[real part|Re]] ''z''}} of each root with its complex conjugate (also a 5th root of unity) is an element of the [[ring (mathematics)|ring]] [[quadratic integer|{{math|'''Z'''[{{sfrac|1&thinsp;+&thinsp;{{sqrt|5}}|2}}]}}]] ({{math|1=''D'' = 5}}). For two pairs of non-real 5th roots of unity these sums are [[multiplicative inverse|inverse]] [[golden ratio]] and [[additive inverse|minus]] golden ratio.

For {{math|1=''n'' = 8}}, for any root of unity  {{math|''z'' + ''{{overline|z}}''}} equals to either 0, ±2, or ±[[square root of 2|{{sqrt|2}}]] ({{math|1=''D'' = 2}}).

For {{math|1=''n'' = 12}}, for any root of unity,  {{math|''z'' + ''{{overline|z}}''}} equals to either 0, ±1, ±2 or ±[[square root of 3|{{sqrt|3}}]] ({{math|1=''D'' = 3}}).