Skip to main content Accessibility help

On linear relations between roots of unity

  • Henry B. Mann (a1)


In a recent paper I. J. Schoenberg [1] considered relations

where the av are rational integers and the ζv are roots of unity. We may in (1) replace all negative coefficients av by −av replacing at the same time ζv by −ζv so that we may, if it is convenient, assume that all av are positive. If we do this and arrange the ζr so that their arguments do not decrease with v then (1) can, as suggested by Schoenberg (oral communication) be interpreted as a convex polygon with integral sides whose angles are rational when measured in degrees. Accordingly we shall call a relation (1) a polygon if all av are non-negative. We shall call a polygon (1) k-sided if all av are positive. The polygon is called degenerate if two of the ζv are equal. Schoenberg calls these polygons polar rational polygons (abbreviated prp) because the vectors composing them have rational coordinates in their polar representations. Schoenberg showed that every prp can be obtained as a linear combination with integral positive or negative coefficients of regular p-gons where p is a prime.



Hide All
1.Schoenberg, I. J., “A note on the cyclotomic polynomial”, Mathematika, 11 (1964), 131136.
2.Waerden, B. L. van der, Modern Algebra, vol. 1. (Frederic Ungar Publishing Co., New York, 1948).
MathJax is a JavaScript display engine for mathematics. For more information see

On linear relations between roots of unity

  • Henry B. Mann (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.