Re: JSH: Resolution now possible

From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 10/17/04 

Date: Sun, 17 Oct 2004 17:00:54 GMT

In sci.math, James Harris
<jstevh@msn.com>
 wrote
on 17 Oct 2004 07:13:23 -0700
<3c65f87.0410170613.507bf41f@posting.google.com>:

For convenience, I will call JSH's "object ring" by the symbol "H".
(J is taken. H probably is too but it's less obvious. :-) )

The standard ring of algebraic integers I'll denote by "A". I'm
not sure if there's a better symbol therefor.

> After over two years of arguing on the specifics of my techniques of
> polynomial factorization which adds to even more years of arguing
> before about factorizations I think it's finally clear how to move to
> resolve all the issues:
>
> 1. My position is that the definition of the ring of algebraic
> integers requiring roots of monic polynomials with integer
> coefficients is arbitrary and misleading in that you can have numbers
> properly units that are excluded on the technicality that they are not
> roots of monic polynomials with integer coefficients.
>
> 2. In support of my position I have given the full algebraic argument
> showing a contradiction between numbers shown to have a specific
> factor, versus exclusion of that factor in the ring of algebraic
> integers based on a technicality.
>
> My work in this area has faced several formal peer reviews and not
> shown to be flawed, though some sci.math'ers have diligently argued
> otherwise and even actively interfered in the journal process by
> sending emails to a journal that had a paper of mine, and succeeded in
> cowing the chief editor Ioannis Argyros so that he withdrew my paper
> without proper cause and without even allowing me to defend against
> the charges the sci.math'ers made.
>
> They broke him completely.
>
> 3. I have outlined a complete ring I call the object ring based on
> two primary requirements:
>
> a. No rational unit other than 1 or -1 is in the ring.

If H contains the algebraic integers as a proper subset one
gets an infinite number of units, of which 2 - sqrt(3),
4 - sqrt(15), and 6 - sqrt(35) are but three examples;
more can be generated at need -- any solution of the
equations x^2 + A * x + 1 and x^2 + A * x - 1 qualify here,
or one can go to higher degrees. (n - sqrt(n^2 - 1)
corresponds to the equation x^2 - 2*n*x + 1.)

Of course +1 and -1 are the only rational units in the ring
of algebraic integers anyway; all other units have to solve
an irreducible equation of the form

x^n + a_{n-1} * x^{n-1} + ... a_1 * x + 1 = 0
or
x^n + a_{n-1} * x^{n-1} + ... a_1 * x - 1 = 0

where n > 1, and no rational number can satisfy such an equation.
(Rational numbers x = p/q satisfy q * x - p = 0. This cannot
be a factor in the above equation unless both p and q are +1 or -1.
One either gets non-integral coefficients or has to deal with
a factor of p or q in the terms for x^n or x^0.)

So you're OK but very close to the edge here.

>
> b. No non-unit member of the ring is a factor of any two integers
> that are coprime in the ring of integers.

Can you more carefully define the term "integer" in both of these contexts?
It appears ambiguous:

[1] No non-unit of H evenly divides two elements of J which are coprime.
[2] No non-unit of H evenly divides two elements of H which are coprime.

with the added proviso that

An element h of H is a unit if another element k in H
exists such that h * k = the multiplicative identity (1).

Since J is Noetherian but A is not, [2] makes absolutely no sense.

http://mathworld.wolfram.com/NoetherianRing.html

>
> Using that definition you can look back at numbers not units in the
> ring of integers and find that they are units in the ring of objects
> which shows how misleading the ring of algebraic integers can be.

Say what?

>
> Basically you can have u_1 u_2 = 1, where u_1 is a unit, but while it
> is in the object ring it's not an algebraic integer because of the
> technicality that it's not the root of a monic polynomial with integer
> coefficients, and that means that u_2 is not a unit.

I would qualify that statement better. This is in H.

So you're stating that u_1 * u_2 = 1 in H, but that one of u_1 or
u_2 is not a unit. This flies in the face of the definition of
a unit; your precipice is cracking...

http://mathworld.wolfram.com/RingUnit.html

[snip for brevity]

>
> http://www.emis.de/journals/SWJPAM/vol2-03.html

The indication on that page is that your submission has been withdrawn.
No explanation on that page is pending.

[rest snipped] 

-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.

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