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

....

[Tim Peters]
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.

[Gene Ward Smith]
I think this has been the problem, or one of them, all along. James
concludes two algebraic integers are coprime, when if you look at his
reasoning that isn't what he ought to be saying. If he'd said of the
three roots of y^3-12*y^2+65=0, one of the roots is a 5-adic unit, then
he would have said something which made sense in terms of the arguments
he was bringing. Instead he says it is coprime to 5, which sends the
whole discussion off in the wrong direction. Moreover, James comes to
this conclusion indirectly, looking at the factorization of the norm
term, and doesn't understand what it is he's really got an example of.

That's right, but he (thinks he) knows what he wants to /conclude/, and that
really drives everything. Arturo gave an excellent account of historical
details, but at the risk of simplifying ;-), the bottom line here is that
James seems to have learned "factorization by inspection" in high school,
never went beyond that, and is endlessly frustrated by that not all rings
"act like" Z wrt factorization.

I'm not sure what he wants from his "object ring", and he doesn't have the
technical vocabulary to explain it, but underlying many of his arguments I
sense anger at rings that don't behave like a unique factorization domain
(or sometimes just an integral domain) when he wants them to. They "block"
him, they have "problems" and "flaws", and mathematicians "lie" when they
say such rings are perfectly fine the way they are.


.

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: My paper, and the cheaters
    ... >> and algebraic integers a and b are two roots of m. ... "Advanced Polynomial Factorization." ... Factorization lemma, Ring of algebraic integers ... Here f is a non unit, non zero algebraic integer coprime to 3 and x, ...
    (sci.math)
  • Re: JSH: The "Published" paper he dosent what you to know about.
    ... -if a and b are coprime they remain coprime in any ... ring that contains the algebraic integers. ... the algebraic integers are not like the evens. ... I expect it's a mistake to assume that James knows what mathematicians mean ...
    (sci.math)
  • Re: JSH: The "Published" paper he dosent what you to know about.
    ... mean when they say two elements of a ring are coprime, ... except) when the ring is Z. ... concludes two algebraic integers are coprime, when if you look at his ... although he has shown no contradiction of the theory of ideals ...
    (sci.math)