Re: JSH: Heart of dispute, number properties

From: Nora Baron (norabaron_at_hotmail.com)
Date: 03/23/05 

Date: 23 Mar 2005 12:11:36 -0800

Randy Poe wrote:
> jst...@msn.com wrote:
> > But I can prove that the second example may be like the first,
except
> > with irrationals, where one root is like the integer root before,
and
> > the other is like the root that is a fraction, but both are
> irrational.
>
> I'd be interested in this proof.
>
> But before that, I'd be interested in a statement of
> what it is you are proving.
>
> What does it mean to be "like the integer root"
> or "like a fraction"?
>

  I will presume to answer on behalf of the Great Sphinx.

  "Like the integer root" means: an algebraic integer.

  "Like a fraction" means: an algebraic number which is not an
algebraic integer. It is worth noting that, given any
algebraic integer A, there exists another algebraic integer B
and a rational integer n, such that

           A = B/n,

i.e., every algebraic number is a quotient of an algebraic integer
and an ordinary integer.

  For a polynomial like 2 x^2 + 5 x + 1, Harris appears to be saying
he can find a ring R which contains the roots of this polynomial and
in which the only units are 1 and -1, and one of roots is an integer
of some kind in this ring, perhaps even an algebraic integer, and other

root is a quotient, i.e., an algebraic number which is not an algebraic
integer. In fact, if 2 x^2 + 5 x + 1 is factored in the form
(a x + 1)*(b x + 1), neither a nor b can be an algebraic integer,
so this polynomial may not be adequate to illustrate what he wants.

  The real question is, why does he want a ring which has the
property that he describes? It is clear now that this is not
a matter of his thinking that the mathematics of algebraic
number theory is *incorrect* - it is instead a matter of personal
taste, i.e., his feeling that polynomials should factor in a different
way in some alternative ring versus how they necessarily factor in the
ring of algebraic numbers. He wants more similarity between reducible
and irreducible polynomials than is allowed for within the ring of
algebraic integers. Why he wants this is not clear.

  Well, actually it is clear. The tail is wagging the dog. He wants
it because otherwise there is no point to his paper "Advanced
Polynomial Factorization". And then, lurking behind APF, is the
mighty Hammer itself with which to destroy the Evil Mathematical
Establishment. If the results of APF hold, then Harris believes
he will have a complete (and short) proof of Fermat's Last Theorem.
That is the Hammer. It is a house of cards. Knowing that his
objection to the way factorization is done with existing algebraic
number theory is just a matter of preference means that there
is really no rigorous basis at all behind APF, and the Hammer
is just a derivative delusion. Do not expect him to recognize
this anytime soon.

  Nora B.

 
> "Like" in what way?
>
> - Randy

Relevant Pages

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