### Tuesday, December 08, 2009

## JSH: How you know, Galois Theory lost, group theory lost?

I can give the mathematical demonstration, but if you refuse to accept mathematical proof, it's kind of hard but I'll put it up again and explain how you have to lie to yourself to not realize this example blows up some things.

It asks you to try and DO something.

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.

If any of you are worth a damn, first thing you should notice is that the functions are not normalized.

Checking at x = 0, gives a^2 + a = 0, so a=0 or a=-1, which means a_1(0) = 0 or -1, and a_2(0) = -1, or 0.

Which means that if you are a well-trained student of physics you will normalize the functions. Just like I did, years ago.

If you normalize the functions then the answer jumps out at you.

One normalization is a_1(x) = b_1(x) - 1, as that gives b_1(0) = 0, and then you can assume a_2(0) = 0, so:

7(175x^2 - 15x + 2) = (5b_1(x) + 2)(5a_2(x) + 7)

with normalized function as b_1(0) = a_2(0) = 0.

And now there's no mystery as there is only one way that 7 gets there now.

Notice that's true even in the field of complex numbers.

So to disbelieve that the 7 multiplied in one particular way is to distrust even the field of complex numbers which gives the same result because even in a field 7*1 = 7.

Divide off the 7 now:

175x^2 - 15x + 2 = (5b_1(x) + 2)(5a_2(x)/7 + 1)

There is NO OTHER WAY. It's trivial math. 7*1 = 7. That is true in the field of complex numbers. It's true in the field of algebraic numbers ,and it's even true in the ring of algebraic integers.

But it is NOT true in the ring of algebraic integers in general that a_2(x) has 7 as a factor.

So you have an easy mathematical proof of a problem with a ring many of you may not remember hearing about until I started going on and on about this issue. But the ring of algebraic integers is a base ring for number theory which was used for the mathematical ideas around the field of algebraic numbers, which is what's used with Galois Theory.

Algebraic numbers are ratios of algebraic integers. i.e. x/y where x and y are algebraic integers, and y is nonzero.

But you don't have to know all the abstractions behind the mathematics many of you probably take for granted, because all you need to know is that math people not only argue with me about the result above, but they killed a freaking mathematical journal that published a paper on the problem.

It is BIG.

Now then, posters can reply to me claiming error when there is none. You can verify the mathematics YOURSELF.

When you see people arguing against what you know has to be correct, get a clue!!!

Now then, what can you do?

Well, you can ask a mathematician to work through the problem. Give them the task to divide off the 7.

Why bother?

Because right now they may believe I'm just one guy, ranting on Usenet, with no one listening.

I think many of you ARE listening, realize the problem, and feel overwhelmed.

So first step is to say: start small. You ask someone to divide off the 7. If mathematicians are too intimidating, pick a mathematical physicist.

Pick a string theorist.

One by one you can around the world remove the delusion from these mathematicians that I'm the only one who knows.

These are proud people. They cherish prestige. They love accolades.

The analogy is to swimming naked: They're swimming naked. Drain the pool.

This result is mathematics and at least then you know it was always true. People make mistakes and can compound them.

The story could have been mathematicians acknowledge a major error in "core" number theory found by an amateur with a degree in physics working at solving math problems as a hobby, but they chose to kill the math journal, and run away.

So now it's up to physics people to go catch them.

It asks you to try and DO something.

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.

If any of you are worth a damn, first thing you should notice is that the functions are not normalized.

Checking at x = 0, gives a^2 + a = 0, so a=0 or a=-1, which means a_1(0) = 0 or -1, and a_2(0) = -1, or 0.

Which means that if you are a well-trained student of physics you will normalize the functions. Just like I did, years ago.

If you normalize the functions then the answer jumps out at you.

One normalization is a_1(x) = b_1(x) - 1, as that gives b_1(0) = 0, and then you can assume a_2(0) = 0, so:

7(175x^2 - 15x + 2) = (5b_1(x) + 2)(5a_2(x) + 7)

with normalized function as b_1(0) = a_2(0) = 0.

And now there's no mystery as there is only one way that 7 gets there now.

Notice that's true even in the field of complex numbers.

So to disbelieve that the 7 multiplied in one particular way is to distrust even the field of complex numbers which gives the same result because even in a field 7*1 = 7.

Divide off the 7 now:

175x^2 - 15x + 2 = (5b_1(x) + 2)(5a_2(x)/7 + 1)

There is NO OTHER WAY. It's trivial math. 7*1 = 7. That is true in the field of complex numbers. It's true in the field of algebraic numbers ,and it's even true in the ring of algebraic integers.

But it is NOT true in the ring of algebraic integers in general that a_2(x) has 7 as a factor.

So you have an easy mathematical proof of a problem with a ring many of you may not remember hearing about until I started going on and on about this issue. But the ring of algebraic integers is a base ring for number theory which was used for the mathematical ideas around the field of algebraic numbers, which is what's used with Galois Theory.

Algebraic numbers are ratios of algebraic integers. i.e. x/y where x and y are algebraic integers, and y is nonzero.

But you don't have to know all the abstractions behind the mathematics many of you probably take for granted, because all you need to know is that math people not only argue with me about the result above, but they killed a freaking mathematical journal that published a paper on the problem.

It is BIG.

Now then, posters can reply to me claiming error when there is none. You can verify the mathematics YOURSELF.

When you see people arguing against what you know has to be correct, get a clue!!!

Now then, what can you do?

Well, you can ask a mathematician to work through the problem. Give them the task to divide off the 7.

Why bother?

Because right now they may believe I'm just one guy, ranting on Usenet, with no one listening.

I think many of you ARE listening, realize the problem, and feel overwhelmed.

So first step is to say: start small. You ask someone to divide off the 7. If mathematicians are too intimidating, pick a mathematical physicist.

Pick a string theorist.

One by one you can around the world remove the delusion from these mathematicians that I'm the only one who knows.

These are proud people. They cherish prestige. They love accolades.

The analogy is to swimming naked: They're swimming naked. Drain the pool.

This result is mathematics and at least then you know it was always true. People make mistakes and can compound them.

The story could have been mathematicians acknowledge a major error in "core" number theory found by an amateur with a degree in physics working at solving math problems as a hobby, but they chose to kill the math journal, and run away.

So now it's up to physics people to go catch them.