# 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:

 $\displaystyle x^7 - 1$ $=$ $\displaystyle \paren {x - 1} \paren {x^6 + x^5 + x^4 + x^3 + x^2 + x + 1}$ $\displaystyle \leadsto \ \$ $\displaystyle \epsilon^6 + \epsilon^5 + \epsilon^4 + \epsilon^3 + \epsilon^2 + \epsilon + 1$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \epsilon^3 + \epsilon^2 + \epsilon + 1 + \epsilon^{-1} + \epsilon^{-2} + \epsilon^{-3}$ $=$ $\displaystyle 0$

But we have:

 $\displaystyle \epsilon$ $=$ $\displaystyle \cos \dfrac {2 \pi} 7 + i \sin \dfrac {2 \pi} 7$ $\displaystyle \epsilon^{-1}$ $=$ $\displaystyle \cos \dfrac {2 \pi} 7 - i \sin \dfrac {2 \pi} 7$ $\displaystyle \leadsto \ \$ $\displaystyle \epsilon + \epsilon^{-1}$ $=$ $\displaystyle 2 c$ where $c = \cos \dfrac {2 \pi} 7$ $\displaystyle \leadsto \ \$ $\displaystyle \epsilon^2 + \epsilon^{-2} + 2$ $=$ $\displaystyle 4 c^2$ squaring $\displaystyle \leadsto \ \$ $\displaystyle \epsilon^3 + \epsilon^{-3} + 3 \paren {\epsilon + \epsilon^{-1} }$ $=$ $\displaystyle 8 c^3$ cubing $\displaystyle \leadsto \ \$ $\displaystyle \epsilon^2 + \epsilon^{-2}$ $=$ $\displaystyle 4 c^2 - 2$ $\displaystyle \leadsto \ \$ $\displaystyle \epsilon^3 + \epsilon^{-3}$ $=$ $\displaystyle 8 c^3 - 6 c$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {8 c^3 - 6 c} + \paren {4 c^2 - 2} + 2 c - 1$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle 8 c^3 + 4 c^2 - 4 c - 1$ $=$ $\displaystyle 0$

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

$x^3 + x^2 - 2 x - 1$ 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.

$\blacksquare$