
Chapter 7
Ruler-and-Compass Constructions
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 ...