Construction of Regular Heptagon by Compass and Straightedge Construction is Impossible

Theorem
There is no compass and straightedge construction for a regular heptagon.

Proof
Construction of a regular heptagon is the equivalent of constructing the point $\tuple {\cos \dfrac {2 \pi} 7, \sin \dfrac {2 \pi} 7}$ from the points $\tuple {0, 0}$ and $\tuple {1, 0}$

Let $\epsilon = \map \exp {\dfrac {2 \pi} 7}$.

Then $\epsilon$ is a root of $x^7 - 1$.

We have:

But we have:

Thus $2 c$ is a root of the polynomial $x^3 + x^2 - 2 x - 1$

But from Irreducible Polynomial: $x^3 + x^2 - 2 x - 1$ in Rationals:
 * $x^3 + x^2 - 2 x - 1$ is irreducible over $\Q$.

Thus by Algebraic Element of Degree 3 is not Element of Field Extension of Degree Power of 2, $\cos \dfrac {2 \pi} 7$ is not an element of any extension of $\Q$ of degree $2^m$.

The result follows from Point in Plane is Constructible iff Coordinates in Extension of Degree Power of 2.