Inconstructible Regular Polygon

Date: 02/22/2002 at 22:18:21
From: Roger Yeh
Subject: Unconstructible regular polygon

Hello Dr. Math,

I've been trying to find a proof that a regular polygon with n sides 
is inconstructible if n is not a Fermat prime number. I know that 
Wantzel (1836) wrote that proof, but have no luck finding it.

Thanks for your time and talent.
Roger Yeh

Date: 02/23/2002 at 00:45:56
From: Doctor Paul
Subject: Re: Unconstructible regular polygon

This is actually not a result of Wantzel. Wantzel proved the 
impossibility of contructing an angle of 20 degrees (and hence that a 
60 degree angle could not be trisected).

It was Gauss who first claimed that a regular n-gon is constructible 
if and only if n = 2^k * p_1 * ... * p_r where k is a non-negative 
integer (possibly zero) the p_i are distinct Fermat primes.

Quoting from Kevin Brown's MathPages:

   Constructing the Heptadecagon
   http://mathpages.com/home/kmath487.htm   

   "Interestingly, although Gauss states in the strongest 
   terms that his criteria for constructibility (based on 
   Fermat primes) is necessary as well as sufficient, he 
   never published a proof of the necessity, nor has any 
   evidence of one ever been found in his papers (according 
   to Buhler's biography)."

I'm not a Gauss historian and I don't know if he had a proof of this 
or not.  The article above would suggest that while he knew it was 
true, he never published a proof.  The reason I think Gauss probably 
did not have a proof is that the proof with which I am familiar uses 
mathematics that was not fully developed and completely understood 
until after Gauss' death.  

I am referring to the branch of mathematics known as Galois Theory.  
Skipping the details (which require some graduate level mathematics to 
understand), the main result of Galois Theory as it relates to 
constructibility of regular polygons is that the regular n-gon can be 
constructed by straightedge and compass if and only if phi(n) (ie, the 
Euler Phi function) is a power of 2. Decomposing n into prime powers 
to compute phi(n) we see that this means

   n = 2^k * p_1 * ... * p_r 

is the product of a power of two and distinct off primes p_i, where 
p_i - 1 is a power of two. It is easy to see that a prime p with 
p-1 a power of two must be of the form p = 2^(2^s) + 1 for some 
integer s. Such primes are called Fermat primes.

You asked for the proof and I've kind of skirted the issue because I'm 
afraid that the you won't understand the mathematics behind it. You 
need to understand terms such as field extension, splitting field, 
separable polynomial, automorphism group, fixed field, and Galois 
group to be able to follow the proof of this fact.  If you really want 
to pursue this futher, I refer you to the 2nd edition of a  nice text 
by Dummit and Foote (the title is _Abstract Algebra_). Chapters 13 and 
14 contain all of the terminology you need and the proof culminates in 
Section 14.5 on page 582. Notice that Section 13.3 contains a proof 
that the three Greek problems from antiquity (trisecting an angle, 
doubling a cube, and squaring a circle) are all impossible. Once the 
necessary machinery has been built up, the impossibility of 
constructing certain n-gons becomes a trivial thing to prove. But it 
is the building up of machinery that is quite difficult.

I hope this helps. Please write back if you'd like to talk about this 
more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/