1 Galois actually proved that you cannot trisect a general angle by intersecting lines and circles. In the Analytic Foundations of Geometry we see that in Theorems 1.4 and 1.6 that the coordinates of new points that one can construct from old points by intersecting lines are rational functions of the coordinates of the old points. In Theorems 3.2 and 3.3, we see that the new coordinates of new points can be computed from the coordinates of the old points by adding only one radical to a rational computation. Theorem 3.13 shows that the situation is similar if we intersect two circles. It can be shown using trigonometry and Galois theory that the coordinates of the point where a line through the origin that makes a 20o angle with the x-axis meets the unit circle cannot be obtained in that manner.

Math 150

Steve Wilson