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From: edp@jareth.enet.dec.com (Eric Postpischil (Always mount a scratch monkey.))
Newsgroups: comp.sys.handhelds
Subject: Re: Finding roots of Tertiary and above equations?
Message-ID: <21932@shlump.nac.dec.com>
Date: 9 Apr 91 18:13:25 GMT
References: <40686@netnews.upenn.edu> <40731@netnews.upenn.edu> <21917@shlump.nac.dec.com> <40762@netnews.upenn.edu>
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Reply-To: edp@jareth.enet.dec.com (Eric Postpischil (Always mount a scratch monkey.))
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In article <40762@netnews.upenn.edu>, hoford@sequoia.circ.upenn.edu
(John Hoford) writes:

>But what is sqrt(2)? The only way I know it is as the solution to x^2=2.

Besides knowing that sqrt(2) is a solution to x^2=2, we also know it is
between 1.4 and 1.5 and that it is irrational.  What does it mean to
"know" something?  Do we know 2.5 better than sqrt(2)?  Part of the
answer is that knowing something means knowing what its properties are,
how it acts, how it relates to other things.  2.5 is 2 plus one-half; it
is half of five.  So we "know" sqrt(2) to the extent that we can square
it, divide it into other numbers, et cetera.  We can work with it, and
that is our knowledge about it.

>I gusss what I am asking is, what does it meen to say something
>has an algebraic soloution and
>is ther any thing realy special about 5'th and above degree
>polynomials.

One meaning of the word "algebraic" is "involving only a finite number
of repetitions of addition, subtraction, multiplication, division,
extraction of roots, and raising to powers" (Webster's New Collegiate
Dictionary).  In that meaning, polynomials of ONLY degree four and less
have algebraic solutions.  Polynomials of greater degree may have
algebraic solutions, as x^8=n obviously does, but this is not
guaranteed.  Some polynomials of degree five and greater do not have
solutions that can be found with only a finite number of repetitions of
those operations.

>What makes the functions sqrt() and "x^(1/y)" ok to use?
>Is this just convention?

Well, in one sense they are okay to use because that's what the word
"algebraic" means.  To answer a bit more deeply, the word "algebraic"
means that because that definition is useful -- those functions are well
understood.  We can implement them with computers or by hand, we can
analyze expressions involving them, et cetera.

In other words, those are basic simple operations.  So, in a sense,
equations that can be solved with those operations are simpler than
equations that cannot, and we call such solutions algebraic solutions. 
It's just a way of giving a name to a class of things.

There is a proof that polynomials of degree five and greater do not have
algebraic solutions, but it involves group theory beyond what I studied.
You might be able to get an answer by asking in sci.math.


				-- edp (Eric Postpischil)
				"Always mount a scratch monkey."
				edp@jareth.enet.dec.com