ABSTRACT
31.1 Some very ancient problems are asking whether certain geometric constructions may be carried out only using ruler and compass. Using coordinates one can relate the points constructible in the plane (see 31.4), by the methods allowed, to the so-called constructible numbers (31.11). Applying only very basic methods, we arrive at the fact that any constructible number is contained in a field extension of the rationals of dimension 2 n for some n (Proposition 31.23). From this we can quickly derive negative solutions to those classic problems: the squaring of the circle, the doubling of the cube in volume, the trisection of an angle. Note that the squaring of the circle is always impossible, the other problems can be solved by more ingenious methods requiring more than ruler and compass. At the same time we have obtained methods to discuss in detail when regular polygons are constructible (31.31). This problem is first reduced to a number of vertices that is a prime power (Proposition 31.34) and then completely solved in Gauss’ Theorem, Theorem 31.39.
