If this was always the case, things would be very simple: Galois theory would just be the study of the “shapes” formed by the roots of polynomials, and the symmetries of those shapes. It would be a branch of planar geometry. But things are not so simple. If we look at the solutions to x^5 ? 2 = 0, something quite different happens:
We will see later on how to obtain these expressions for the roots. A pentagon has 10 geometric symmetries, and you can check that all arise as symmetries of the roots of x^5 ? 2 using the same reasoning as in the previous example. But this reasoning also gives a symmetry that moves the vertices of the pentagon according to:
This is not a geometrical symmetry! Later we will see that for p > 2 a prime number, the solutions to x^p ? 2 = 0 have p(p ? 1) symmetries. While agreeing with the six obtained for x^3 ? 2 = 0, it gives twenty for x5 ? 2 = 0. In fact, it was a bit of a fluke that all the number theoretic symmetries were also geometric ones for x^3 ?2 = 0. A p-gon has 2p geometrical symmetries and 2p ? p(p?1) with equality only when p = 3. （つづく）
