NetNews Usenet Archive 1992 #19/NN_1992_19.iso/spool/sci/math/10911

Text File | 1992-09-02 | 2.7 KB | 50 lines

Newsgroups: sci.math
Path: sparky!uunet!gatech!news.byu.edu!news.mtholyoke.edu!nic.umass.edu!cis.umassd.edu!ursa.cis.umassd.edu!martin
From: [e-mail] (Gary Martin)
Subject: Re: Abel's proof of the insolubility of the quintic
In-Reply-To: [e-mail]'s message of Wed, 2 Sep 1992 20:42:29 GMT
Message-ID: <[e-mail]>
Sender: [e-mail] (USENET News System)
Organization: University of Massachusetts Dartmouth
References: <[e-mail]>
Date: Thu, 3 Sep 1992 02:27:31 GMT
Lines: 37

In article <[e-mail]> [e-mail] writes:

   Does anyone know a reference for a modern treatment of Abel's
   proof of the insolubility of the quintic?  A high-school student
   recently came to me, having learnt the solutions to the lower degree
   equations, and hoping to understand the higher degree case as well.
   He tried to read Abel's memoir (translated into English in the venerable
   Source book in Mathematics).  Understandably, he didn't get too far
   since Abel's paper is rather dense and obscure to an untutored reader.
   (For instance he doesn't take any pains to explain what the meaning
   of `a solution in radicals to the general quintic' would actually mean.)

   One suggestion to this student is to read enough about groups and fields
   to learn the rudiments of Galois Theory.  There is a book of H. Edwards
   which more or less travels this route.  On the other hand, Abel's proof
   is quite elementary, and can in principle be explained to a high-school
   student.  (I said in principle!)  I am looking for a source for such an
   explanation; any pointers would be greatly appreciated.

I don't know a good reference, but as I recall, you need to know a
little bit about field extensions, how groups permute roots of
polynomials, the relation between solvability of groups and solvability
of equations, and the fact that the alternating group of degree 5 is
simple (and hence that the corresponding symmetric group is not
solvable).  I remember sketching the proof in about an hour for a
very sharp senior engineering major at Swarthmore about 7 years ago
using Fraliegh's algebra text as a guide.  The student was just about
to begin a first semester algebra course, so we had to go through the
necessary definitions.  I think we probably started with the chapter
in which the insolvability is proved and traced backwards through the
definitions and theorems as we discovered which ones we needed.  It
turned out not to be very much, and as I said, he was very sharp, so
it worked well.

--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
[e-mail]