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From: Amy on 2 Oct 2009 12:11

"enry" <invalid(a)invalid.com> wrote in message
news:ha3br7$a4b$1(a)news.albasani.net...
> There is a 3-page postscript document on EMIS.
>
> Here is the PDF version.
>
> http://dl.free.fr/gMojEGqSw/jsh.pdf
>
> (note: this "paper" was never printed in paper, and was "withdrawn")
>
> ........................................................................
> Electronic Journal: Southwest Journal of Pure and Applied Mathematics
> Internet: http://rattler.cameron.edu/swjpam.html
> ISSN 1083-0464
>
> Issue 2, December 2003, pp. 6-8.
> Submitted: July 25, 2003. Published: December 31 2003.
> ADVANCED POLYNOMIAL FACTORIZATION
> James Harris
>
> Abstract. Algebraic method for determining distribution of factors within
> a polynomial
> factorization, which breaks through what was seen as a barrier from
> overinterpretations
> of Galois Theory.
>
> A.M.S. (MOS) Subject Classification Codes. 11R04,11R09
>
> KeyWords and Phrases. Polynomial factorization, Galois theory,
> Factorization
> lemma, Ring of algebraic integers
>
> Advanced Polynomial Factorization Approached.
>
> Determining the distribution of factors within irrational algebraic
> integers has
> long been considered impossible as it is not possible to do using Galois
> Theory.
> However a simple technique through the introduction of more variables
> makes it
> possible. To highlight the standard belief consider the algebraic integer
> roots of
> x2 + x ? 5.
>
> While you know that the algebraic integer factors are themselves factors
> of 5,
> can either not have non unit factors of 5? How do you know?
>
> In looking to consider distribution of algebraic integer factors within a
> factorization
> I'll be using a more complicated example than x2 + x ? 5.
>
> This paper will show, using basic algebraic methods, that given the
> factorization,
> in the ring of algebraic integers,
>
> 65x3 ? 12x + 1 = (a1x + 1)(a2x + 1)(a3x + 1)
>
> one of the a's is coprime to 5.
>
> First I'll need a simple lemma to generalize beyond factors of a
> polynomial that
> are themselves polynomials.
>
> E-mail Address: jstevh(a)msn.com
> c 2003 Cameron University
>
> Typeset by AMS-TEX
>
> 6
> ADVANCED POLYNOMIAL FACTORIZATION 7
> Factorization Lemma:.
>
> Given a factor g of a polynomial P(x), further defined as a factor for all
> x, which
> means that the value of g for a value 'a' of x is a factor of P(a), within
> the ring of
> algebraic integers, there exists r and c such that
>
> g = r + c
>
> where r=0, or varies as x varies, and c is a factor of the constant term
> P(0) and
> is itself constant.
>
> Let x=0, then g must be a factor of P(0), so at that point c = g.
>
> If when x does not equal 0, g=c, r=0. If when x does not equal 0, g 6=c
> there
> must exist r which varies with x. That is, r=g-c. 
>
> As an example consider ?x + 1 which is a non polynomial factor of x+1, and
> while there are an infinity of irrational solutions consider the rational
> solution at
> x=35.
>
> Then I have ?35 + 1 = 6 = 5 + 1; therefore when x=35, g=6, r=5, and c=1.
> But for different values of x, g and r will vary, while c will not.
>
> Primary Argument.
>
> Given
>
> 65x3 ? 12x + 1 = (a1x + 1)(a2x + 1)(a3x + 1)
>
> in the ring of algebraic integers. Let
>
> P(m) = f2((m3f4 ? 3m2f2 + 3m)x3 ? 3(?1 + mf2)xu2 + u3f)
>
> Here f is a non unit, non zero algebraic integer coprime to 3 and x, and u
> a non
> unit, non zero algebraic integer coprime to f. Note P(m) has a factor that
> is f2.
>
> That expression comes from expanding (v3+1)x3?3vxy2+y3, using the
> substitutions
> v = ?1+mf2, and y = uf, where additional variables provide an additional
> degree of freedom.
>
> Now consider the factorization
>
> P(m) = (a1x + uf)(a2x + uf)(a3x + uf)
>
> where multiplying out shows that
>
> a1a2a3 = m3f6 ? 3m2f4 + 3mf2 = f2(m3f4 ? 3m2f2 + 3m)
> so
>
> a1a2a3 = mf2(m2f4 ? 3mf2 + 3).
>
> Therefore, at least one of the a's cannot be coprime to m, and at least
> one of
> the a's must equal 0 when m=0.
>
> (Note: The a's are roots of a monic polynomial with algebraic integer
> coefficients
> so they are algebraic integers.)
>
> Notice that the constant term P(0) is given by P(0) = f2(3xu2 + u3f) and
> also
> that P(0)/f2 = 3xu2 + u3f, which is coprime to f.
>
> 8 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS
> Then I have the factor of P(m), g1, where g1 = a1x+uf, where here I also
> have
> that a1 is not coprime to m.
>
> �From my factorization lemma, I have that, when m=0
> g1 = c = uf
>
> meaning f is a factor of the constant term.
>
> Therefore, exactly two of the a's equal 0, when m=0, to get the factor f2
> in the
> constant term P(0), while one must not equal 0, or f3 would be the factor.
>
> Now as noted before in general P(m) has a factor that is f2, and
> separating that
> factor off, gives a constant term coprime to f; therefore, given
>
> g1 = a1x + uf where with m = 0, g1 gives a factor of f it must have that
> same
> factor in general, proving that two of the a's have a factor that is f.
> Therefore, one factor is coprime to f.
>
> Now letting m=1, f=?5, where I can let u=1 as its value doesn't change the
> a's,
> I have
>
> (m3f6 ? 3m2f4 + 3m)x3 ? 3(?1 + mf2)xu2 + u3 = 65x3 ? 12x + 1
> which may be more easily seen from using v = ?1 + mf2 = 4, y=1 with (v3 +
> 1)x3 ? 3vxy2 + y3.
>
> Therefore, with the factorization
>
> 65x3 ? 12x + 1 = (a1x + 1)(a2x + 1)(a3x + 1)
>
> one of the a's is coprime to 5, which shows where some of the algebraic
> integer
> factors distribute despite the factors being irrational.
>
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>
> and that is it folks! Finally, there it is.
> Even JSH his own self would not reveal the contents
> It will cause world economy to collapse,
> Iran attacked
> Ants to jump!
>
>

Seems JSH's main point is;


Factorization Lemma:.

Given a factor g of a polynomial P(x), further defined as a factor for all
x, which means that the value of g for a value 'a' of x is a factor of
P(a), within the ring of algebraic integers, there exists r and c such
that

g = r + c

where r = 0, or varies as x varies, and c is a factor of the constant term
P(0) and is itself constant.


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