Galois Theory, 5th Edition

Chapter 21Circle Division

DOI: 10.1201/9781003213949-21

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

$$\begin{array}{l}{\zeta }_{11}=\cos \frac{2\pi }{11}+\text{i}\sin \frac{2\pi }{11}\hfill \end{array}$$

look like?

As the theory stands, the best we can offer is

$$\begin{array}{l}\sqrt[11]{1}\hfill \end{array}$$(21.1)

which is not terribly satisfactory, because the obvious interpretation of $\sqrt[11]{1}$ is 1, not ζ₁₁. Gauss's theory of the 17-gon hints that there might be a more impressive answer. ...