Galois Theory, 5th Edition

Chapter 7Ruler-and-Compass Constructions

DOI: 10.1201/9781003213949-7

Already we are in a position to see some payoff. The degree of a field extension is a surprisingly powerful tool. Even before we get into Galois theory proper, we can apply the degree to a warm-up problem—indeed, several. The problems come from classical Greek geometry, and we will do something much more interesting and difficult than solving them. We will prove that no solutions exist, subject to certain technical conditions on the permitted methods.

According to Plato the only ‘perfect’ geometric figures are the straight line and the circle. In the most widely known parts of ancient Greek geometry, this belief had the effect of restricting the (conceptual) instruments ...