Doubling a Cube

"Doubling a cube" is also known as Delian Problem.  The story behind this famous problem is as follows: About 430 BC a great plague stuck Athens.  The Athenian consulted the oracle at Delos to find a way to stop the plague.  The oracle said the plague occurred because Apollo was angry.  The only way to stop it was to double the volume of Apollo's altar, which had the form of a cube.  Athenian tried to double the length, height and breadth of the altar but obviously volume was not doubled, but eight times larger.  This made Apollo even angrier and the plague persisted until about 423 BC. The following is ﻿the Temple of the Delians at Delos, dedicated to Apollo. (credit: www.achtypistours.gr)
It is easy to see that the length of the side of the required cube is the product of and the length of the side of the original one.  Therefore, Delian problem boils down to determining whether  is a constructible number.   We will show that it is in fact not constructible i.e. it is impossible to double a cube by straightedge and compass.  By the Main Theorem, it suffices to show that the degree of over is not a power of . First of all, satisfies the cubic equation Then it is not hard to see that is irreducible over . Therefore, the degree of over is i.e. not a power of . This completes the proof of impossibility of doubling a cube. Then we will see how Greek mathematicians solved this problem by other tools.