Friday, September 22, 2006
JSH: Simple explanation, coverage problem
Having noted that it is unlikely that any number theorist will acknowledge the coverage problem of the ring of algebraic integers on their own—as it invalidates their degrees—I might as well explain, yet again, just how easy it all is, and why there is this problem.
Remember, algebraic integers are defined to be roots of monic polynomials with integer coefficients.
That definition can be proven to require that NONE of the roots of a monic polynomial with integer coefficients that is irreducible over Q can be coprime to any prime factor of the last coefficient.
It is a quirk of the ring, and mathematicians wrongly concluded over a hundred years ago that meant that non-rationals were somehow special in comparison to rationals, so that you had this odd and special result.
How special?
Well, look at some simple quadratic examples:
x^2 - 3x + 2 = (x-2)(x-1)
is a pairing of a rational unit with a rational non-unit in the ring of algebraic integers.
THAT CANNOT HAPPEN with non-rationals in the same ring.
So mathematicians assume that with the roots of something like
x^2 - 5x + 2 = 0
where you do not have rationals, that 2 is somehow split up, and factors between the two roots.
But what my research proves is that nope, it doesn't have to split up (though it still might) as the ring of algebraic integers will simply NOT allow either root to have 2 as a factor IN THAT RING.
That does not mean, however, that 2 is not a factor of only one root while the other is a unit.
Confused? Consider evens with 2 and 6, and note that in evens 2 does not have a factor in common with 6 because 3 IS NOT EVEN.
Because 2 can't be a factor in the ring of algebraic integer with my simple irreducible quadratic example of one root, the ring of algebraic integers just does not allow the other root to be a unit. Easy. But it still can actually be a unit.
So how do you know? Turns out that you can prove it with some careful mathematical analysis and a special tool I call non-polynomial factorization. Factor a polynomial into non-polynomial factors and you can see all kinds of neat and weird things, as well as RIGOROUSLY prove the problem.
I have a published paper that says I accomplished the proof, and a retraction of that paper, and the later death of the journal to let you understand how big the politics here are.
The math is easy. The consequences are what's hard.
Proving that the ring of algebraic integers has this coverage problem requires that you factor polynomials into non-polynomial factors, which is the new thing that I brought to the table, which showed the problem.
The problem hasn't been proving that the techniques I use work, as the proof is easy relying on mostly trivial algebra, but in climbing over the waves of denial that have swept the mathematical world, or maybe I should say the Usenet math world, though I have had contacts off of Usenet.
It's easy to prove the coverage problem, but if number theorists accept it, they lose any results that depend on the flawed ideas, and that is a huge thing, as it can be shown that it takes away the theory of ideals, and changes how Galois Theory is seen to work.
So how does it change the view on how Galois Theory works?
Simplest answer is that taking away the supposed difference between rational and non-rational numbers means that Galois Theory tells you only as much with non-rationals, as it does with rationals.
Number theorists have shown that they will not accept the truth on their own.
And it's not a big surprise why they will not, and why others when they realize the scope of the problem turn tail and run, because, for instance, Wiles did not prove Taniyama-Shimura, which is then still a conjecture, as the theory of ideals and the old way of using Galois Theory just went away.
With this result, people like Wiles may have no major proofs discovered at all in their entire careers and with the truth, may not be able to justify their positions and status.
So, if Wiles acknowledged this result, next thing he'd need to do would be to step down from any positions that would come from thinking the flawed results were correct, as would mathematicians around the world.
Understand the problem?
It's not about mathematical proof but about human denial of mathematical proof, where the people with the power would lose that power by telling the truth, by invalidating their own expertise. It's a freaky thing that I guess has never happened in the world's intellectual history, where the very people who are tasked with acknowledging a result are the ones who would lose their expert status by doing so.
[A reply to someone who said that any mathematician that would find an inconsistence in the definition of algebraic integers would become famous and therefore would not hide it.]
The problem is the definition leads to a ring with a coverage problem.
And no, they wouldn't be famous—they'd be infamous among all the mathematicians invalidated by the result!
Like, let's say some relatively unknown mathematician at some university takes this result and runs with it, down the road, it topples Wiles, Ribet and Taylor, who end up with NO MAJOR RESULTS for their entire careers.
That mathematician would have to run the gauntlet with just about every other number theorist in the world gunning for them.
Before it were over that person would be called every name in the book, have people attacking personally, and even maybe trying to probe into personal information—all in a growing campaign to shoot that person down, no matter what.
Nope. Math people don't have those kind of balls. The reaction to this result that I've seen is terror or denial, like with that math grad student at Cornell University.
Rather than being my poster boy for total fear in the graduate community, he could have been out there pushing the result, but by running and hiding, he can still try for a career as a mathematician.
You know, he does read sci.math, as he's responded to me a few times about ripping on him.
But abject terror defines the math community in this area, or total denial.
None of you have the courage to push the truth here.
Remember, algebraic integers are defined to be roots of monic polynomials with integer coefficients.
That definition can be proven to require that NONE of the roots of a monic polynomial with integer coefficients that is irreducible over Q can be coprime to any prime factor of the last coefficient.
It is a quirk of the ring, and mathematicians wrongly concluded over a hundred years ago that meant that non-rationals were somehow special in comparison to rationals, so that you had this odd and special result.
How special?
Well, look at some simple quadratic examples:
x^2 - 3x + 2 = (x-2)(x-1)
is a pairing of a rational unit with a rational non-unit in the ring of algebraic integers.
THAT CANNOT HAPPEN with non-rationals in the same ring.
So mathematicians assume that with the roots of something like
x^2 - 5x + 2 = 0
where you do not have rationals, that 2 is somehow split up, and factors between the two roots.
But what my research proves is that nope, it doesn't have to split up (though it still might) as the ring of algebraic integers will simply NOT allow either root to have 2 as a factor IN THAT RING.
That does not mean, however, that 2 is not a factor of only one root while the other is a unit.
Confused? Consider evens with 2 and 6, and note that in evens 2 does not have a factor in common with 6 because 3 IS NOT EVEN.
Because 2 can't be a factor in the ring of algebraic integer with my simple irreducible quadratic example of one root, the ring of algebraic integers just does not allow the other root to be a unit. Easy. But it still can actually be a unit.
So how do you know? Turns out that you can prove it with some careful mathematical analysis and a special tool I call non-polynomial factorization. Factor a polynomial into non-polynomial factors and you can see all kinds of neat and weird things, as well as RIGOROUSLY prove the problem.
I have a published paper that says I accomplished the proof, and a retraction of that paper, and the later death of the journal to let you understand how big the politics here are.
The math is easy. The consequences are what's hard.
Proving that the ring of algebraic integers has this coverage problem requires that you factor polynomials into non-polynomial factors, which is the new thing that I brought to the table, which showed the problem.
The problem hasn't been proving that the techniques I use work, as the proof is easy relying on mostly trivial algebra, but in climbing over the waves of denial that have swept the mathematical world, or maybe I should say the Usenet math world, though I have had contacts off of Usenet.
It's easy to prove the coverage problem, but if number theorists accept it, they lose any results that depend on the flawed ideas, and that is a huge thing, as it can be shown that it takes away the theory of ideals, and changes how Galois Theory is seen to work.
So how does it change the view on how Galois Theory works?
Simplest answer is that taking away the supposed difference between rational and non-rational numbers means that Galois Theory tells you only as much with non-rationals, as it does with rationals.
Number theorists have shown that they will not accept the truth on their own.
And it's not a big surprise why they will not, and why others when they realize the scope of the problem turn tail and run, because, for instance, Wiles did not prove Taniyama-Shimura, which is then still a conjecture, as the theory of ideals and the old way of using Galois Theory just went away.
With this result, people like Wiles may have no major proofs discovered at all in their entire careers and with the truth, may not be able to justify their positions and status.
So, if Wiles acknowledged this result, next thing he'd need to do would be to step down from any positions that would come from thinking the flawed results were correct, as would mathematicians around the world.
Understand the problem?
It's not about mathematical proof but about human denial of mathematical proof, where the people with the power would lose that power by telling the truth, by invalidating their own expertise. It's a freaky thing that I guess has never happened in the world's intellectual history, where the very people who are tasked with acknowledging a result are the ones who would lose their expert status by doing so.
[A reply to someone who said that any mathematician that would find an inconsistence in the definition of algebraic integers would become famous and therefore would not hide it.]
The problem is the definition leads to a ring with a coverage problem.
And no, they wouldn't be famous—they'd be infamous among all the mathematicians invalidated by the result!
Like, let's say some relatively unknown mathematician at some university takes this result and runs with it, down the road, it topples Wiles, Ribet and Taylor, who end up with NO MAJOR RESULTS for their entire careers.
That mathematician would have to run the gauntlet with just about every other number theorist in the world gunning for them.
Before it were over that person would be called every name in the book, have people attacking personally, and even maybe trying to probe into personal information—all in a growing campaign to shoot that person down, no matter what.
Nope. Math people don't have those kind of balls. The reaction to this result that I've seen is terror or denial, like with that math grad student at Cornell University.
Rather than being my poster boy for total fear in the graduate community, he could have been out there pushing the result, but by running and hiding, he can still try for a career as a mathematician.
You know, he does read sci.math, as he's responded to me a few times about ripping on him.
But abject terror defines the math community in this area, or total denial.
None of you have the courage to push the truth here.
# posted by James Harris @ 20:51 
