Theorem of Lindemann

A real number is said to be transcendental over  if it is not algebraic over  i.e. it is not a solution to any polynomial equation with coefficients in .  After thousand years of futile attempt to square a circle, more and more mathematicians tended to believe that it is impossible and conjectured that  is transcendental over .  Since any constructible number is algebraic over , the conjecture implies that it is impossible to square a circle by Euclidean construction. In 1882, Ferdinand von Lindemann (1852-1939), a German mathematician, finally proved this conjecture and hence completely settled the third classical problem in Greek geometry.
The proof of the conjecture involves heavy use of complex numbers and calculus.  For interested students, the following is the link to the proof: In fact, there is nothing in the proof that is too advanced for an undergraduate who is major in mathematics.  However, time and patience is needed to go through the proof because it is rather long and complicated.