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### Inconstructible Regular Polygon

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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:

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.

more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Constructions
College Modern Algebra
College Number Theory
College Triangles and Other Polygons
High School Constructions
High School Number Theory
High School Triangles and Other Polygons

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