Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site petsd.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!mhuxr!mhuxt!houxm!vax135!petsd!cjh
From: cjh@petsd.UUCP (Chris Henrich)
Newsgroups: net.math
Subject: Re: roots for quintic polynomial
Message-ID: <537@petsd.UUCP>
Date: Mon, 10-Jun-85 12:52:12 EDT
Article-I.D.: petsd.537
Posted: Mon Jun 10 12:52:12 1985
Date-Received: Tue, 11-Jun-85 04:47:41 EDT
References: <2206@utcsstat.UUCP>, <412@linus.UUCP>
Organization: Perkin-Elmer DSG, Tinton Falls, N.J.
Lines: 37

[]
	Robert D. Silverman writes:

> Polynomials of degree less than or equal to 4 always have a Galois
> group which is of prime order and hence cyclic and therefore
> solvable. That is why there are general solutions for these.

I don't think this is correct.  The Galois group of an
equation of order n is a subgroup of the group S/sub n/ of
permutations on n things.  For n <= 4 this group is solvable;
hence the Galois group of an equation of order <= 4 is
solvable.  But it needn't be cyclic.

A theory exists for the solution of equations of order 5 in
"closed" form where the form involves an elliptic modular
function and its inverse.  This function plays a role in the
solution analogous to the role of the exponential (and its
inverse, the logarithm) in computing n-th roots for the
"closed form" solutions of quadratics, cubics, and quartics.
There is a book by Felix Klein, called _Lectures_On_The_
Icosahedron_, which used to be in print (Dover paperback) in
an English translation.  It's beautiful mathematics, if you
happen to find it beautiful, but far from being a practical
algorithm.

Can anyone give a modern reference for the algorithmic
aspects of computing Galois groups, and using them to deduce
properties of equations?

Regards,
Chris

--
Full-Name:  Christopher J. Henrich
UUCP:       ..!(cornell | ariel | ukc | houxz)!vax135!petsd!cjh
US Mail:    MS 313; Perkin-Elmer; 106 Apple St; Tinton Falls, NJ 07724
Phone:      (201) 758-7288