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 look like? As the theory stands, the best we can offer is 11√1, which is not terribly satisfactory, because the obvious interpretation is 1, not ζ11. Gauss's theory of the 17-gon hints that there might be a more impressive answer. 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 explain why and how.
