
Chapter 21
Circle Division
To halt the story of regular polygons at the stage of ruler-and-compass constructions
would leave a small but significant gap in our understanding of the solution of poly-
nomial equations by radicals. Our definition of ‘radical extension’ involves a slight
cheat, which becomes evident if we ask what the expression of a root of unity looks
like. Specifically, what does the radical expression of the primitive 11th root of unity
ζ
11
= cos
2π
11
+ i sin
2π
11
look like?
As the theory stands, the best we can offer is
11
√
1 (21.1)
which is not terribly satisfactory, because the obvious interpretation of
11
√
1 is 1, not
ζ
11
. Gauss’s theory of