Re: Attacking my algebraic integer work

jstevh_at_msn.com
Date: 03/29/05 

Date: 28 Mar 2005 16:01:08 -0800

mhochster@gmail.com wrote:
> jstevh@msn.com wrote:
> > mhochster@gmail.com wrote:
> > > jstevh@msn.com wrote:
> > > > It seems to me that the position that mathematicians made a
> mistake
> > > > over a century ago is a GREAT one to go after very vigorously
as
> in
> > > > many ways it is my most dramatic claim.
> > > >
> > >
> > > A good start would be a clear statement of what the mistake is.
If
> > the
> > > mathematicians made a mistake, they made it in their own words.
> Let's
> > > see a citation followed by an explanation of why the conventional
> > > wisdom is wrong.
> >
> > Good suggestion. I don't have a citation.
> >
> > I have said for some time that there was a mistake over a hundred
> years
> > ago, and my take on that is that I've simply gone back to when
> > algebraic integers were introduced, which was over a hundred years
> ago.
> >
> > I haven't actually gone and looked at texts from then, though I'm
> sure
> > I have seen some excerpts posted by someone else on the sci.math
> > newsgroup.
> >
> > As for an explanation of what I've said the problem is, I think
> trying
> > that yet again would be counter-productive.
> >
> > Putting up yet another explanation with the same old claims is just
> > more self-promotion, and what I'd prefer to do is attack the
> > mathematical argument that forms the basis for my claim, which I
> admit
> > I haven't yet given in this thread.
> >
>
> But while you are in self-critical mode, you should also consider the
> possibility that mathematicians do not actually hold the false belief
> you are attributing to them. A clear statement of the alleged error
> from an authoritative source would at least lay that question to
rest.

Let's see. Consider a ring where the following two properties hold:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n
such that mn is an integer, and if mn is not a factor of m, then n
cannot be a unit in the ring.

Those properties hold with the ring of integers, the ring of gaussian
integers, and the ring of algebraic integers.

But, it cannot be shown that ANY of those rings exhaust all
possibilities.

That is, maybe there is some other ring, where those properties hold
that has members that are not integer, gaussian integers, nor algebraic
integers.

If so, then that means that you might have an algebraic numbers x/y
where x and y are members of that ring, but not algebraic integers.

But you may know that every algebraic number can be written as a ratio
of coprime algebraic integers, so what if that number can be written as

a/b

where a and b are algebraic integers, so you have

a/b = x/y

where in the more inclusive ring, you have

cx/cy = a/b

where cx and cy are algebraic integers?

Then coprimeness in that ring would not mean coprimeness in the more
inclusive ring.

But if you don't realize that possibility, you can have an odd thing,
where you can prove "coprimeness" by relying on coprimeness in the ring
of algebraic integers, and algebraically find that two numbers are not
coprime, and thus have the appearance of proving two different and
opposite things.

A simple analogy that I've given before is to consider 6 and 2 in the
ring of evens.

Then you can have 6/2 coprime in that ring, but, of course, in the ring
of integers, that is just 3.

Does that make sense?

Can you see why having a bigger ring that includes the ring of
algebraic integers, but has members outside of it where those key
properties hold, would cause problems?

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)