ABSTRACT
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 polynomial 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
ζ11 = cos 2pi 11
+ isin 2pi 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 ζ11. Gauss’s theory of the 17-gon hints that there might be a more impressive answer. In place of 17
√ 1 Gauss has a marvellously complicated system of nested square roots,
which we repeat from equation (20.9):
cos 2pi 17
= 1 16
( −1+
√ 17+
√ 34−2
√ 17
+
√ 68+12
√ 17−16
√ 34+2
√ 17− 2(1−
√ 17) √
34−2 √
17 )
with a similar expression for sin 2pi17 , and hence an even more impressive formula for ζ17 = cos 2pi17 + isin
Can something similar be done for the 11th root of unity? For all roots of unity? The answer to both questions is ‘yes’, and we are getting the history back to front, because Gauss gave that answer as part of his work on the 17-gon. Indeed, Vandermonde came very close to the same answer 25 years earlier, in 1771, and in particular he managed to find an expression by radicals for ζ11 that is less disappointing than (21.1). He, in turn, built on the epic investigations of Lagrange.
