Re: New paper, algebraic integers, Galois Theory

From: James Harris (jstevh_at_msn.com)
Date: 10/17/04 

Date: 17 Oct 2004 15:56:16 -0700

"W. Dale Hall" <mailtowd-hall@pacbell.net> wrote in message news:<8cycd.14748$nj.8215@newssvr13.news.prodigy.com>...
> James Harris wrote:
> > jstevh@msn.com (James Harris) wrote in message news:<3c65f87.0410160547.1cb5e61@posting.google.com>...
> >
> >>"W. Dale Hall" <mailtowd-hall@pacbell.net> wrote in message news:<8UZbd.14310$nj.12329@newssvr13.news.prodigy.com>...
> >>
> >>>James Harris wrote:
> >>>
> >>>
> >>>>"W. Dale Hall" <mailtowd-hall@pacbell.net> wrote in message news:<v8Mbd.13602$nj.3896@newssvr13.news.prodigy.com>...
> >
> >
> > <deleted>
> >
> >>> >>>
> >>> >>> Two algebraic integers u and v so that a_i and 5 are shown to be
> >>> >>> coprime in the ring of algebraic integers. No one can deny the
> >>> >>> truth then, bucko. Here it is:
> >>> >>>
> >>> >>> a_i u + 5 v = 1.
> >>> >>>
> >>> >>> Just fill in the u and v, and
> >>> >>
> >>> >>
> >>> >>
> >>> >> Not necessary, as I *prove* they exist.
> >
> >
> > OOPS! I don't.
> >
> >
> >>> >
> >>> >
> >>> > You're claiming that the common factor (I've called it r, I believe)
> >>> > is a unit, but not a unit in the algebraic integers. I, on the
> >>> > other hand, have the equations
> >>> >
> >>> > a_i = qr
> >>> > 5 = rs
> >>> >
> >>> > for q,r,s in the ring of algebraic integers. For a_i and 5 to be
> >>> > coprime, this leads to
> >>> >
> >>> > (qr) u + (rs) v = 1,
> >>> >
> >>
> >>Why?
> >
> >
> > And that's a pertinent question, which goes back to the entire issue
> > of relying on a result true in the ring of algebraic integers, as if
> > that's all that matters.
> >
> > But it's a technicality, which represents one of the more interesting
> > areas where people can fail to understand mathematics.
> >
> >
> > James Harris
>
> Please. You've claimed that the numbers a_i and 5 are coprime in the
> ring of algebraic integers. To my knowledge, no one has ever seriously
> claimed that that ring is all that matters. For instance, there is the
> issue of global poverty. The inability (whether real or perceived) of
> subjugated peoples to obtain what they consider justice is also of real
> concern. Even within mathematics, no one seriously considers the ring
> of algebraic integers to be "all that matters". Narrowing down to
> algebraic number theory, the role of algebraic integers looms pretty
> large, but *still* no one would say they're "all that matters", since
> having that field of fractions around is mighty useful.
>
> But that's no what you meant, was it? No, you probably meant in the
> context of your assertion that the numbers a_i and 5 are coprime in
> the ring of algebraic integers. In that context, well, yes. If you're
> going to say that a_i and 5 are coprime *in that ring*, then *that ring*
> is all that matters.
>
> Now, your correction, in which you now state that in this situation,
> which merely restates the definition of coprimeness in the particular
> situation at hand:
>
> a_i and 5 are coprime in the ring of algebraic integers
> if and only if there exist algebraic integers u and v
> for which
>
> u a_i + v 5 = 1
>
> you now *NO LONGER CLAIM* to have proven that u and v exist.

Well, yeah, as the assertion is equivalent to claiming that there
exists an algebraic integer that is the root of a non-monic polynomial
with integer coefficients, which it can't be.
 
> Therefore, you *NO LONGER CLAIM* that a_i and 5 are coprime in the
> ring of algebraic integers. The coprimeness claim is one and the same
> as the claim that the u and v exist as algebraic integers.

No it's not.

Imagine that 3 is arbitarily not allowed to be a factor in some
special ring.

Now someone keeps arguing with you that 9 is prime, because it's prime
in that ring, but you know that 9 = 3(3), and say so.

Ultimately it's settled down to them admitting they are making a
special claim based on their special, arbitrary rule.
 
> Do we agree?
> Yes? Then your paper was wrong.
> No? Then where is the disagreement?
>
> Dale

The arbitrary requirement is that algebraic integers are roots of a
monic polynomial with integer coefficients.

The paper *proves* coprimeness.

Your objection is no different than if some people decided to get
together and claim that 3 is not a factor, so that they could have 9
prime.

Like maybe it's a cult of 9, so they wish 9 to be prime, so they say 3
is not a factor in their special ring.

Arbitrarily selecting out numbers that are not roots of a monic
polynomial with integer coefficients is simply a more subtle error,
leading to what appears to be a contradiction in that you can prove
coprimeness--as I do in my paper--and then someone like yourself can
come back and say by some arbitrary rule that it doesn't exist:
contradiction.

The proof of the error is in the contradiction in that the paper is
algebraically correct.

Challenges to it depend on some arbitrary rule, which if accepted over
the algebra makes math inconsistent.

It's not complicated.

If my paper has an error, then show an error in the paper, and don't
come back later to say the argument IS correct but some arbitrary rule
means the paper is not correct.

James Harris

Relevant Pages

  • Re: My paper, and the cheaters
    ... > coprimeness result to handle the problem with the ring of algebraic ... > the problem with the ring of algebraic integers. ... we need to discuss the Galois theory ...
    (sci.math)
  • Re: JSH: Critique means slow, and thorough
    ... coprimeness in the more inclusive ring. ... can have an odd thing, where you can prove "coprimeness" by relying on coprimeness in the ring of algebraic integers, and algebraically ... particular field extension called the Hilbert class field of F. ...
    (sci.math)
  • Re: Math rules and publication
    ... > even faced formal peer review. ... > out of the ring of algebraic integers has to be determined separately. ... a piece of purported mathematics is accepted or not, ...
    (sci.math)
  • Re: Inconsistency with algebraic integers
    ... with the ring of algebraic integers as using it you can appear to ... is NOT a factor of EITHER root in the ring of algebraic integers!!! ... I use identities throughout much of the paper, ... Him telling me I'm not a real mathematician is about as significant as my ...
    (sci.math)
  • Re: JSH: Inconsistency with algebraic integers
    ... in the ring of algebraic integers. ... That is formally a factorization. ... Mathematics abhors inconsistency. ...
    (sci.math)