14. Geometric Constructions

In this chapter, we discuss some straightedge-and-compass constructions. We are particularly interested in the following classical problems: the impossibility of squaring the circle, doubling the cube and angle trisection. The problems of impossibility (or possibility) of some geometric constructions by using different means (such as straightedge-and-compass, only compass, or other means) are usually discussed in courses on Galois theory. Meanwhile, the classical impossibility problems mentioned above do not require extensive knowledge on Galois groups (if any at all); only some basic knowledge of finite field extensions is required. Other problems, such as Gauss’ theorem on straightedge-and-compass constructible regular polygons require further knowledge related to Galois groups of field extensions. This chapter contains several exercises concerned with geometric straightedge-and-compass constructions. We prove two theorems: the first tends to be used in proofs of impossibility of some straightedge-and-compass constructions; the second tends to be used in proofs of possibility.