Inconstructible Regular PolygonDate: 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/ |
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