From: wcw@math.psu.edu (William C Waterhouse)
Newsgroups: sci.math
Subject: Re: solvable quintics ??
Date: 13 Sep 1996 21:33:29 GMT
In article gno@selene.rz.uni-duesseldorf.de,
rosentha@classic41.rz.uni-duesseldorf.de () writes:
> .... by Dummitt ?
> Sometime ago i posted a question regarding solvable quintics and was promptly
> answered. I received word of an article by "Dummitt" on solvable quintics
> en detail in a journal named 'computation and math'. Sadly I was not able to
> locate any of the above mentioned data....
Here's the information (taken from mathscinews, the on-line version of
Mathematical Reviews):
Dummit, D. S. Solving solvable quintics, Mathematics of Computation
57 (1991), 387-401.
Corrigenda, Math. Comp. 59 (1992), 309.
William C. Waterhouse
Penn State
==============================================================================
From: Herman Bruyninckx
Newsgroups: sci.math.num-analysis
Subject: Re: Closed Form Root of Quintic Poly
Date: Wed, 4 Nov 1998 08:00:45 +0100
> > Am I wasting time searching for a closed form solution to a degree
> > 5 polynomial. I have found refereces for cubic and quartic (if I am
> > reading things correctly).
> >
> > Is there a closed form solution for a quintic polynomial ???
> >
Take a look at:
@book{King96,
author = {King, R. Bruce},
title = {Beyond the quartic equation},
year = {1996},
publisher = {Birkhauser},
address = {Boston, MA},
issn_isbn = {0-8176-3776-1}
}
@article{KingCanfield91,
author = {King, R. Bruce and Canfield, E. Rodney},
title = {An algorithm for calculating the roots of a general
quintic equation from its coefficients},
journal = {Journal of Mathematical Physics},
year = {1991},
volume = {32},
pages = {823--825}
}
@book{Shurman97,
author = {Shurman, Jerry M.},
title = {Geometry of the Quintic},
year = {1997},
publisher = {John Wiley \& Sons},
issn_isbn = {0471130176}
}
@article{Stillwell95,
author = {Stillwell, John},
title = {Eisenstein's Footnote},
journal = {The Mathematical Intelligencer},
year = {1995},
volume = {17},
number = {2},
pages = {58--62}
}
--
Herman.Bruyninckx@mech.kuleuven.ac.be (Ph.D.) Fax: +32-(0)16-32 29 87
Dept. Mechanical Eng., Div. PMA, Katholieke Universiteit Leuven, Belgium
==============================================================================
From: Paul Abbott
Newsgroups: sci.math.num-analysis
Subject: Re: Closed Form Root of Quintic Poly
Date: Thu, 05 Nov 1998 11:04:50 +0800
[partial quote of above post deleted -- djr]
Also, you might want to take a look at
http://www.wolfram.com/posters/quintic/
Not only is the poster very nice there are a good set of references for
this problem here.
Cheers,
Paul
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA 6907 mailto:paul@physics.uwa.edu.au
AUSTRALIA http://www.physics.uwa.edu.au/~paul
God IS a weakly left-handed dice player
____________________________________________________________________
==============================================================================
From: Robin Chapman
Newsgroups: sci.math
Subject: Re: Insolubility of quintic without Galois Theory
Date: Thu, 26 Nov 1998 08:21:20 GMT
In article <73iaqs$t1v$1@nnrp1.dejanews.com>,
torquemada@my-dejanews.com wrote:
> From the Mathematica web site:
>
> Finally, Ruffini (1799) and Abel (1826) showed that the solution of
> the general quintic cannot be written as
> a finite formula involving only the four arithmetic operations and the
> extraction of roots. By 1832 Galois
> had developed the theory of Galois groups and described exactly when a
> polynomial equation is solvable.
>
> How did Ruffini and Abel show this without Galois theory? Or had they really
> discovered the rudiments of Galois theory? According to
> http://www-history.mcs.st-and.ac.uk/~history/Mathematicians/Ruffini.html
> Ruffini's proof was later called the Wantzel modification of Abel's proof.
> Roughly what methods did this use?
>
In Heinrich D"orrie's 100 Great Problems of Elementary Mathematics
(Dover) there's a proof that an irreducible soluble polynomial of odd
prime degree p over Q has either 1 real root or p real roots. This proof
doesn't use any finite group theory. Perhaps this was the method that
Abel & Ruffini used?
Robin Chapman + "They did not have proper
SCHOOL OF MATHEMATICal Sciences - palms at home in Exeter."
University of Exeter, EX4 4QE, UK +
rjc@maths.exeter.ac.uk - Peter Carey,
http://www.maths.ex.ac.uk/~rjc/rjc.html + Oscar and Lucinda, chapter 20
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