Saturday, 23 April 2011

Constructructable Polygons, and X^17 = 1

A while back I wrote a blog about Gauss' proof that the 17 gon (heptadecagon) was constructable.  Dave Renfro then set himself the goal of writing a clear and complete explanation such that a really good high school student (or more particularly, a somewhat slow high school teacher like me) could understand.  After very many hours of labor and multiple "final" drafts he has done a super job. It is a document well worth the attention of every upper level HS math student and teacher.  Thanks, Dave.

  I have posted his complete document. A Detailed and Elementary Solution to x^17=1  ,
and below is his introduction.


This manuscript was written during a three week period in March-April 2011 and it was motivated by Pat Ballew's 20 March 2011 blog entry Gauss and Constuctable Polygons <http://pballew.blogspot.com/>. It occurred to me that, in all of the many dozens of expositions about solutions to x^17 = 1 I had encountered in 30+ years, none gave all the details for obtaining one of the lengthy square root expressions one often sees exhibited. This includes the huge number of class notes and unpublished manuscripts that have populated the internet in the past 15 years, although that may no long be the case after this manuscript circulates. I suppose a case could be made that this is done in Klein [44] (Article 6, pp. 29-32) and in Barnard/Child [4] (pp. 174-175). However, I have not encountered any treatment where sufficient detail was given so that someone competent in high school algebra and trigonometry, but otherwise not very mathematically sophisticated, could be reasonably expected to follow. With this in mind, there are three main ways that I have tried to distinguish the present treatment from the many already in existence.

First, I have tried to be extremely detailed and explicit in the exposition, especially in carrying out algebraic manipulations, in rewriting numerical expressions, and in giving precise explanations for how the manipulations and rewriting are done.

Second, in the case of obtaining explicit square root expressions related to the equation x^17 = 1, no computer algebra systems or calculators are used. In fact, I believe the only instance in which I needed to multiply something out by hand using grade school arithmetic methods was (34)(170) = 578 at one point, although even this could have been avoided, as will be evident when I obtain a more compact square root expression for Cos(2 Pi/17) .

Third, I have paid as much attention to the completeness and accuracy of the bibliography as I have with the mathematics. Thus, author names, journal titles, page citations, and the like are detailed and explicit. I have also given careful thought to the items included, with the exception of the x^257=1 entries (whose novelty and rarity merit the exception). My intended audience consists mainly of English readers who are high school students and their teachers, faculty at small colleges, and other math enthusiasts who do not have access to the resources of a research university library (an extensive book and journal collection, JSTOR and other expensive digital collections, etc.). Thus, the items in the bibliography were mostly selected for being freely available on the internet and for assuming relatively little mathematical background knowledge. I made a few exceptions to these principles, but this was done when I thought an item might be especially useful to someone interested in exploring the subject further. Also, I have noted which books have been reprinted by Dover Publications, because such a reprinting (even if no longer in print) makes it much more likely that the book can be found on the shelves of a public library or a small college library.
 
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