Saturday, December 23, 2006

 

JSH: Reasoning behind the hostility

For years I have wondered why there was so much anger from math people where none of the explanations made sense, until now with this remarkably simple and perfect argument that among other things proves that just one root of

x^2 - 6x + 35 = 0

has 7 as a factor in the ring of algebraic integers.

Why would so many mathematicians get upset with such a result?

Well, when Hilbert was talking about the great problems in mathematics, if mathematicians were being honest they could have added finding how factors of 7 distribute among roots of a simple quadratic like that to the list.

But compare that to the 10 problems he did give and you can get some sense of why a proud group of people would rather run from an annoying reality that would escape most people because of point of view, as, after all, we have the quadratic formula, and lots of methods for APPROXIMATING the solution for the roots.

You need such in science, construction and practical areas, but in number theory, approximate is no solution at all.

To understand how little the quadratic formula actually tells you without evaluating the square roots consider the roots of

x^2 -12x + 35 = 0

from

x = (12 +/- sqrt(4))/2

where I don't evaluate the square root though it's trivial, so you can understand how little you actually know there, until you evaluate the square root.

But with x^2 - 6x + 35 = 0, you have

x = (6 +/- sqrt(-104))/2

and while you can divide out that 2 to get

x = 3 +/- sqrt(-26)

at that point you're stuck, and how does 7 distribute?

No one knew, so they made stuff up.

What I did was figure out an easy way to figure out that one root has 7 as a factor, while it still doesn't say which root, but it gets around actually evaluating that square root with some simple analytical tools, and there's the problem.

Now the truth would come out about the previous refusal to acknowledge a hard problem in a seemingly trivial area, where mathematicians made up a solution which doesn't actually work.

For instance, you may think that a non-monic primitive polynomial with integer coefficients irreducible over Q cannot have algebraic integer roots, when it can.

The fake argument actually proves that a monic primitive with integer coefficients irreducible over Q cannot have the root of a monic primitive with integer coefficients of lesser degree, and then there is one of those annoying leaps when people use what's true to leap to something that's not.

So it turns out that to have an algebraic integer root, a non-monic primitive with integer coefficients irreducible over Q must have the root of a monic primitive with integer coefficients irreducible over Q of HIGHER DEGREE.

Go back, check arguments you thought were proofs see the obvious and realize that people when pushed can lie—even in mathematics.

So no, Hilbert wouldn't have wanted to state that one of the great unknowns in mathematics was how 7 or its factors distributed with the roots of

x^2 - 6x + 35 = 0

and with the faked math, he didn't have to, but instead could put up really complicated and highly technical stuff that made mathematicians look like the brilliant people they want people to think they are—not people still working at what most people think of as trivially solved because they don't understand number theory.

In number theory, approximate is no solution at all.

[A reply to someone who said that James has alredy been exposed to three proofs of the fact that, given an equation like x² − 6x + 35 = 0, then both roots either are, or are not, divisible by 7 in the algebraic integers.]

You're wrong.

I have a perfect argument, simple proof which readers can see at my
Extreme Mathematics group:

http://groups.google.com/group/extrememathematics

What I did was settle another area where people seemed to be puzzled, as I showed how you get that neat little way of factoring into non-polynomial factors that I do.

By generalizing a bit I also give a final solution for the a's where you can actually check to see if it's possible for one of the a's to NOT have 7 as a factor and see the consequences if that possibility were to occur.

So I've added more detail and added more flash to the paper, before it goes to the next journal.

The next chief editor won't have the option, like the Bulletin one did, of saying the paper focuses on too narrow of an area to interest readers!

It is a beautiful argument. A wonderful proof so I'm glad I had to find it versus just having my previous line of attack shown in my old paper.

And it's so wild to be able to drop the negatives against Dedekind as the ring of algebraic integers is ok. He did good.





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