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Path: utzoo!linus!bs
From: bs@linus.UUCP (Robert D. Silverman)
Newsgroups: net.math
Subject: Re: roots for quintic polynomial
Message-ID: <412@linus.UUCP>
Date: Thu, 6-Jun-85 10:08:05 EDT
Article-I.D.: linus.412
Posted: Thu Jun  6 10:08:05 1985
Date-Received: Sun, 9-Jun-85 22:16:51 EDT
References: <2206@utcsstat.UUCP>
Organization: The MITRE Coporation, Bedford, MA
Lines: 41

> Subject: roots of quintic polynomial
> 
> 	I am looking for equations (and references, if possible) which can be
> used to determine the roots of a fifth-order polynomial. Any help would be
> greatly appreciated !!!
> -- 
> 
>        	{allegra,ihnp4,linus,decvax}!utzoo!utcsstat!anthony
>         {ihnp4|decvax|utzoo|utcsrgv}!utcs!utzoo!utcsstat!anthony
 
Organization: The Mitre Corporation, Bedford MA
In article <2206@utcsstat.UUCP> you write:
>Subject: roots of quintic polynomial
>
>	I am looking for equations (and references, if possible) which can be
>used to determine the roots of a fifth-order polynomial. Any help would be
>greatly appreciated !!!
>-- 
>
>       	{allegra,ihnp4,linus,decvax}!utzoo!utcsstat!anthony
>        {ihnp4|decvax|utzoo|utcsrgv}!utcs!utzoo!utcsstat!anthony


I hate to be the bearer of unhappy news but no such general solution
of a quintic exists. A general polynomial of 5th degree or higher 
(in fact any polynomial) is solvable in terms of radicals only when
its Galois group is solvable. This is an extremely rare event for high
order polynomials.

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.

Basically, the only way to solve high order polynomials is either:
 
	(1) numerically
	(2) trigonometrically (i.e. via the substitution x --> exp(i theta))
	    this can lead to a solution in terms of trig functions of algebraic
	    angles, but it is not always clear how to solve the resulting
	    equation.