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

jstevh@xxxxxxx wrote:
marcus_b wrote:
jstevh@xxxxxxx wrote:
marcus_b wrote:

<deleted>

One of the simpler cases to which your result should
apply is when f = sqrt(5). In that case, the polynomial
equation above becomes

a^3 + 12 a^2 - 65 = 0.

This is a monic polynomial with integer coefficients.
It is irreducible over the rationals. The roots
are algebraic integers. Dedekind's theorem
and Galois theory both say that none of the roots are
coprime to f^2 = 5 in the ring of algebraic integers.
You however infer that one of the roots IS coprime
to 5. And since you are saying that Galois theory is
wrong, you must be saying that it is wrong in the ring
to which it applies: the ring of algebraic integers.
From which we must conclude: you think one of roots is
coprime to 5 *** in the ring of algebraic integers.***
This is inescapable.

Nope.

Why go in circles? I already note that the result isn't true in the
ring of algebraic integers, so why come back and claim that I'm saying
it's true in the ring of algebraic integers?

It's trivial math too.

You're doing what other posters have tried to do which is claim I'm
saying what I'm not, and denying the reality that I acknowledge that
none of the roots can be coprime to 5 in the ring of algebraic
integers.

Your paper says Galois theory disagrees with your algebra.
The Galois theory in question applies to the ring of algebraic
integers, not to arbitrary rings. Therefore you must be saying
that your results - particularly your claim that one of a1(m),
a2(m) or a3(m) is coprime to f - must apply in the ring of
algebraic integers. Otherwise you have no reason to say
Galois theory is incorrect.

Marcus.


It's false implication.

One can correctly say that 2 is coprime to 6 in evens, right?

But what if some mathematician proclaims that means that 2 is coprime
to 6 in a larger sense?

No one would, of course, because you know that 2*3 = 6.

But what I've done is prove a similiar situation where it's not so
trivial to point to the error, so mathematicians who if they admit the
error have no accomplishments in their entire careers can dance around
the proper conclusion.

There's no error! If there was, why wasn't it discovered LONG before then? Why isn't anyone running to your defense?

So that means that Galois Theory doesn't say anything differently for
non-rationals than it does for rationals as the ring of algebraic
integers is irrelevant.

And where did you learn Galois Theory?

The ring of algebraic integers is a historical oddity, with no
mathematical importance.

Taking the roots of monic polynomials with integer coefficients doesn't
have any real import in a mathematical sense, but is just some quirky
idea some people thought might be useful---and it wasn't.


James Harris


Dave
.

Relevant Pages

  • Re: JSH: The "Published" paper he dosent what you to know about.
    ... the "Over Iinterpertations of Galois Theory" by mathematicians. ... Determining the distribution of factors within irrational algebraic integers ... one of the a's is coprime to 5. ... that are themselves polynomials. ...
    (sci.math)
  • 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)
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