Compass and Straightedge Construction for Regular Heptagon does not exist/Proof 2

From ProofWiki
Jump to navigation Jump to search

Theorem

There exists no compass and straightedge construction for the 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:

\(\ds x^7 - 1\) \(=\) \(\ds \paren {x - 1} \paren {x^6 + x^5 + x^4 + x^3 + x^2 + x + 1}\)
\(\ds \leadsto \ \ \) \(\ds \epsilon^6 + \epsilon^5 + \epsilon^4 + \epsilon^3 + \epsilon^2 + \epsilon + 1\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \epsilon^3 + \epsilon^2 + \epsilon + 1 + \epsilon^{-1} + \epsilon^{-2} + \epsilon^{-3}\) \(=\) \(\ds 0\)

But we have:

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

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.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Compass_and_Straightedge_Construction_for_Regular_Heptagon_does_not_exist/Proof_2&oldid=583193"