Thursday, April 06, 2006


JSH: So now it's easy

As people have argued with me I've continued to simplify and abstract, and now it has come down to a rather simple statement—though odd in particular ways—about factoring a polynomial:

7*C(x) = (f(x) + 7)*(g(x) + 1) where f(0) = g(0) = 0

in the complex plane where C(x) is a polynomial, and you have its factorization on the right, with functions that are 0 at x=0.

Add in the obvious logical point that the distributive property is uncaring about the value of the internals of the group being multiplied and you have the result that can be used to show that the theory of ideals doesn't work.

And that's why people keep arguing with me about it.

The distributive property is simply a statement that if you multiply a group, you multiply the elements within that group, so when 7 is multiplied times the polynomial C(x), it multiplies times the internal elements of C(x).

Being a clever person, I split up C(x) internally into two main factors, and ask the question, how does 7 or its factors distribute?

Using the logic that the value of elements within a group can't affect the distributive property, it's trivial to prove that 7 can only have multiplied through one of those two main factors of C(x).

Why is it a big deal?

Because you can take the general result—true on the complex plane—go to the ring of algebraic integers and get a contradiction!!!

So the ring of algebraic integers contradicts with a result valid over the complex plane, oh, and also valid in the ring of integers.

That proves the ring of algebraic integers has special properties previously unknown, which is why the result is generally invalid in that ring.

Considering the issues carefully that leads to the conclusion that ideal theory must not work.

It's such a trivial argument at the start that it is hard to believe that it leads to showing such complicated mathematical ideas as those used in ideal theory to be flawed, but that's how the wrong ideas stay in place.

Remember, I am the one here arguing who has a peer reviewed and published result in this area, though yes, sci.math'ers emailed the editors of the journal and managed to get my paper yanked, but it still went through formal peer review, and it still was published.

The mathematics is trivial to the point of disbelief that people could argue about it, or that it could shoot down such massive and long-held ideas.

And yes, mathematicians are just people too so they can sit on important information, hold on to ideas that have been shown wrong, just as easily as people held on to the idea that the earth was the center of the universe for so long.

You might say that ideal theory is the center of these people's universe and they refuse to let it go, even though it is so easily, and trivially, proven to be false.

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