Ruler and compass constructions

Abstract

Abstract The last three chapters have been concerned primarily with finite fields. In this chapter we turn our attention to infinite fields and in particular to subfields of the real numbers ℝ. Our interest in such fields is derived from a construction problem in Euclidean geometry. Many geometric constructions can be performed using only a pencil, compass, and unmarked ruler. For instance, suppose we wish to bisect an angle θ, as in Fig. 5.1. This can be done, using our limited tools, as follows. Firstly use the compass to draw a segment of a circle cutting the sides of the angle at points A and B say. Then draw two circles of equal radius centred at A and B; these two circles intersect at points C and D, say, as in Fig. 5.2. The line CD bisects θ.