Three Greek problems of antiquity
The three Greek problems of antiquity were some of the most famous unsolved problems in history. They were first posed by the Greeks but were not settled till the advent of Abstract algebra and Analysis in modern times.
All three constructions have been shown to be impossible.
The impossibility of these constructions relies on the characterization of the constructible numbers. It can be shown that, starting with a segment of length , one can construct a segment of length iff can be formed from the rational numbers in a finite number of steps using only the operations addition, subtraction, multiplication, division, and taking square roots. Using field theory, one can then show that must be an algebraic number and that the degree of it's minimal polynomial must be a power of .
Trisection of the General Angle
Statement: Given an angle, construct by means of straight edge and compass only, an angle one-third its measure. This is impossible because the number would have to be constructable in this case, and the minimal polynomial of is , which has degree .
Doubling of a Cube
Given a cube, construct by means of straight edge and compass only, a cube with double the volume. This is impossible, because the number would have to be constructed, and the minimal polynomial of is , which has degree .
Squaring of a Circle
This is impossible, as the transcendental number would have to be constructable in this case.