Sunday, December 06, 2009


JSH: Core error and Galois Theory

A few years back I did my own amateur research in what I thought was abstract number theory, and even briefly got published before some sci.math people ruined that with some emails. But funny thing. The ENTIRE math journal then died.

Later I simplified the argument now to a seemingly simple task that I give, but pulling on the thread I've pointed out that the "core error" is not just some abstract problem for number theorists as it removes the usefulness of Galois Theory. And because group theory comes from Galois Theory it brings into question, group theory.

The math part is easy:

Try to divide off the 7:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x) + 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0 in the ring of algebraic integers.

The 7 on the left hand side looks like a trivial factor. Standard teaching is just to divide off such routine factors, for instance:

7(x^2 + 3x + 2) = (7x + 7)(x + 2)

becomes x^2 + 3x + 2 = (x + 1)(x + 2), as the factor of 7, is useless, and provides no important information.

The example I've given above in many ways is just as trivial, the 7 is just as extraneous and it's removed just as easily as:

175x^2 - 15x + 2 = (5a_1(x)/7 + 1)(5a_2(x) + 7)


175x^2 - 15x + 2 = (5a_1(x) + 7)(5a_2(x)/7 + 1)

as it implies that one of the a's has 7 itself as a factor, but that means that:

a^2 - (7x-1)a + (49x^2 - 14x) = 0

has 7 as a factor of only one root, and if you let x=1, you get a^2 - 6a + 35 = 0, and

a_1(1) = 3 +/- sqrt(-26), or a_2(1) = 3 -/+ sqrt(-26) or vice versa.

So now you know that one of those has 7 as a factor, though it's invisible because of the square root.

But that blows up a lot of algebraic number theory and that's where all the arguing starts and the dead math journal and lots of unpleasantness and namecalling often ensues.

The problem is that with:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x) + 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0

it's insane to claim that the 7 is a function of x. It's a 7. 7 is just a number. It's NOT a function. Also you KNOW that excess factors can just be divided off. The value of x is irrelevant, but the number theory that blows up actually teaches that how the 7 divides off in the ring of algebraic integers depends on the value of x.

That algebraic integer theory formed the basis of the mathematical ideas underpinning the field of algebraic numbers which is about ratios of algebraic integers, and you get Galois Theory from the field of algebraic numbers.

Remember: dead math journal, years of arguing, math people claiming that 7 divides off as a function, weird things.

If you lose group theory then yeah, that brings into question the Standard Model. So the "core error" if you pull the string really goes places, which is fascinating in and of itself.

Anyone wish to lay odds on mathematicians and physicists losing so many pet cows?

Yeah, so now you see the problem.

Math is easy.

You have to do weird things to believe in the old stuff. The new stuff works great and fits with simple things taught to kids about extra factors. I can explain it all. Easy. Even got published!—until they killed the entire goddamn journal AFTER my paper was pulled.

Isn't that overkill?

Why trash the math journal AFTER they managed to get my paper pulled?

If I'm right then the wrong math can only go so far and problems will arise, but if these people are too invested, they'll just make up stuff, call you a crackpot if you cross them, and do wrong things until something breaks, I guess.

That's the way it goes.

But if you know the math you know to just let them go on their way and waste their lives without investing your precious time in stupidity.

I know I do. I don't even bother reading newspaper reports about physics research any more. Waste of time.

It's not really research any more when people are willfully in error. They're just playing a game.

And a stupid one too.

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