Constructible number
Impossible constructions

The ancient Greeks thought that certain problems of straightedge and compass construction they could not solve were simply obstinate, not unsolvable. However, the non-constructibility of certain numbers proves that these constructions are logically impossible to perform. (The problems themselves, however, are solvable using methods that go beyond the constraint of working only with straightedge and compass, and the Greeks knew how to solve them in this way. One such example is Archimedes' Neusis construction solution of the problem of Angle trisection.)

In particular, the algebraic formulation of constructible numbers leads to a proof of the impossibility of the following construction problems:
punctuation:no_comma