Re: JSH: The "Published" paper he dosen't what you to know about.


Tim Peters wrote:
[William Hughes, to JSH]
Have you still not understood that for the algebraic integers

-if a and b are coprime they remain coprime in any
ring that contains the algebraic integers.

No, the algebraic integers are not like the evens. If something
is coprime in the algebraic integers it stays coprime.

[The converse is not true. If a and b are not coprime in the algebraic
integers it is trivial to find a ring containing the algebraic integers
in which a and b are coprime]

I expect it's a mistake to assume that James knows what mathematicians mean
when they say two elements of a ring are coprime, except (and only except)
when the ring is Z. In the case of Z, he understands a consequence of the
general ring definition as it happens to work out in Z (hardly an accident,
since the general definition is a generalization of what happens to be true
in Z ...), but I bet he isn't even aware of the general definition. Even if
he is, he certainly doesn't know how to apply it.

Think about it ;-) Has he ever said something demonstrating the slightest
idea of what an ideal is, apart from mechanically repeating that he's proved
ideal theory wrong -- and even then, only after someone else gave him the
phrase "ideal theory" to begin with?


There is a *long* history here. James uses the definition that
a and b are coprime if they have no common factor other
than a unit.. He has been told countless times that this is
not the standard definition but ignores this.
[It does not matter much. In most of the rings James uses,
the two definitions are equivalent]

This leads James to the following intuitive result.

If a and b are coprime in a ring you can find a larger
ring in which they are not coprime by adding a common
factor.

He illustrates this by considering the ring of evens. He has been
told, but does not seem to understand, that things are different
in a ring with identity.

An illustration of this can be seen in one of his earlier definitons
of the "object ring". He included a condition that "the object ring"
had the property that it contained the integers and if a and
b were coprime in the integers they would remain coprime in the
object ring. He was told immediately that this was an empty condition,
but it took him years to acknowledge this.

Anyway, I expect that James will shortly realize that he needs
to go from non-coprime pairs to corprime pairs,
not the other way around,
and drop the whole thing (perhaps complaining about
red herrings from mathematicians)

Interestingly, the whold claim that "ideal theory" is flawed started
when
Jame's nose was rubbed in the fact that Dedekind had shown
that the algebraic integers form a Bezout domain. So he may
claim he has shown the result "if a and b are coprime in the
algebraic integers they are coprime in any ring containing the
algebraic integers" to be false and that any proof to the contrary
relies on flaws in "ideal theory". (James promised more details
on these putative flaws years ago, but to date nothing is
forthcomming).

For that matter, I expect it's a mistake even to assume that James
understands what "ring" means. One reason he can't follow obvious
refutations of his goofiest claims appears to be that he literally can't
comprehend what he's reading: if it's concise & accurate, he dismisses at
once as "math-ese", but if it's "folksy" he automatically substitutes in his
own private (& hopelessly careless) meanings for technical phrases. Ask him
to define one of his terms -- his "definition" of "the object ring" is one
of his /better/ <brrrr> efforts of that kind.


To James a definition if one of two things

- A magic spell that will make his proofs work
(credit to Arturo Magidin)

- A distraction from evil mathematicians who are trying
to hide the truth.

-William Hughes

.

Relevant Pages

  • Re: JSH: The "Published" paper he dosent what you to know about.
    ... I'm not sure what he wants from his "object ring", ... so the other factors needed to be coprime to f --- in the ring ... happen in the ring of algebraic integers. ... Dedekind's work and the theory of ideals. ...
    (sci.math)
  • Re: Attacking my algebraic integer work
    ... > The paper Advanced Polynomial Factorization has been retired. ... >> his arithmetic in the ring of algebraic integers. ... not coprime to 5 in some yet-to-be-well-defined ring of objects, ...
    (sci.math)
  • Re: JSH: Brainstorming over, for now
    ... You requested a proof that one could prove that an algebraic number could be expressed as the quotient of two coprime algebraic integers. ... >> So consider what follows from the ring of algebraic numbers, ... You're right, factorization can be done by infants using ordinary household materials, but we're really save, and the NSA doesn't need to get in on the game? ...
    (sci.math)
  • Re: JSH: The "Published" paper he dosent what you to know about.
    ... are algebraic integers. ... You however infer that one of the roots IS coprime ... And since you are saying that Galois theory is ... you must be saying that it is wrong in the ring ...
    (sci.math)
  • Re: Dumb arguments, and social stuff
    ... Actually in the ring of algebraic integers one of the a's IS ... > I can explain any supposed counterexamples, but what I can't do is ... This post contains James' best attempt so far to explain ...
    (sci.math)