NetNews Usenet Archive 1992 #19/NN_1992_19.iso/spool/sci/math/10935

Internet Message Format | 1992-09-03 | 1.9 KB

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From: [e-mail] (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: Abel's proof of the insolubility of the quintic
Message-ID: <[e-mail]>
Date: 3 Sep 92 15:28:32 GMT
Article-I.D.: netnews.87834
References: <[e-mail]> <[e-mail]>
Sender: [e-mail]
Reply-To: [e-mail] (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
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In-reply-to: [e-mail] (Gary Martin)

In article <[e-mail]>, martin@lyra (Gary Martin) writes:
>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).

Actually, it's neat to prove non-solvability of A_n, n>=5 directly: if
we compute right-to-left, (abc)(cde)(cba)(edc)=(adc).  So the commutator
subgroup contains all three cycles, hence all of A_n.  For the proposed
unravel-the-Galois-proof project, this may allow a dispensation with
any explicit group theory.  [This proof is from Artin's little book.]

I spent my undergraduate days under the misconception that the proof of
the simplicity of A_5 was overly complicated and uninformative.  Then I
came across the proof in Rotman: identify the conjugacy classes of A_5,
and notice the impossibility of any of them forming a partition for a
size that non-trivially divides 60.  This too might mean something in
the context of the proposed project.
--
-Matthew P Wiener ([e-mail])