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From: seroussi@hplred.HP.COM (Gadiel Seroussi)
Newsgroups: comp.sys.handhelds
Subject: Re: Finding roots of Tertiary and above equations?
Message-ID: <6230006@hplred.HP.COM>
Date: 19 Apr 91 16:29:18 GMT
References: <25744@hydra.gatech.EDU>
Organization: Hewlett Packard Labs, Palo Alto CA
Lines: 23

In article
/ hplred:comp.sys.handhelds / edp@jareth.enet.dec.com (Always mount a scratch monkey.) /  8:18 am  Apr 17, 1991 /
Eric Postpischil writes

>In regard to previous messages, "algebraic solution" did not refer to the
>numbers that were zeroes of the polynomial being algebraic numbers; it referred
>to the solution itself being algebraic.  In that context, "algebraic" means
					  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>composed of only finite numbers of additions, subtractions, multiplications,
>divisions, raisings to powers, and extractions of roots.
>
>And to answer the previous questions, yes, there is something special about
>polynomials that are quintic and above:  Their solutions cannot be written
>as algebraic (in the sense I have given above) formulae of their coefficients.
>Quartic polynomials and below can be solved with finite numbers of the listed
>operations; quintic polynomials and above cannot, in the general case.

However, the word "algebraic" has a well defined meaning in mathematics 
(especially when talking about roots of polynomials), and the concept
we are referring to here is precisely captured by the term
"solvable by radicals". There is no need to start redefining standard 
mathematical terms.
Gadiel Seroussi