JSH: Galois Theory problem and resolution in complex plane

I found a problem with some old mathematics which takes out the usefulness of Galois Theory, but it's hard to get people to believe that unless they try to do the math, so I gave a challenge to give to a mathematician which should be simple as it's removing an excess factor of 7. Now I'm sure some of you would love a nifty little mathematical challenge with which you could confound your colleagues, and I especially like to pick on string theorists, but first you need to BELIEVE it will, so here's why.

First the challenge.

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.

To understand what is happening you need to go to the complex plane. Next you need normalized functions.

With x=0, you know you zero one of the functions as then:

a^2 + a = 0, so one is 0, and the other is -1. That trivially works:

7( 2) = (5(0) + 7)(5(-1) + 7)

where I arbitrarily picked a_1(0) = 0, a_2(0) = -1, as of course it could be vice versa. You're halfway there.

Remember we're in the complex plane. Now to normalize I can leave one of the a's and add 1 to the other, so, let:

b_2(x) = a_2(x) + 1, so a_2(x) = b_2(x) - 1, then:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5b_2(x) + 2)

and you know that the 7 could only have multiplied times the first factor—even though it's the complex plane.

Why? Because you see the 7 there with the normalized functions. 7*1 = 1.

If it had multiplied times the other factor then you'd have:

7(175x^2 - 15x + 2) = (5a_1(x) + 1)(5b_2(x) + 14)

Because 7*2 = 14, even on the complex plane.

Yup, I picked the first way. I MADE this example up to show a problem, and yes, with an extraneous factor like 7, it takes human choice to force it to go one way, as like: 7(x^2 + 3x + 2) = (7x + 7)(x + 2), there is a CHOICE.

I simply used an example that blows up some faulty math ideas.

So now you know from the complex plane how the multiplication had to go, so what's the big deal?

Well, now you know that only one of the roots of:

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

can actually have 7 as a factor! Trying x=1, you find that a^2 - 6a + 35 = 0, and

a_1(x) = 3 +/- sqrt(-26), and a_2(x) = 3 -/+ sqrt(-26)

or vice versa. But you cannot SEE the factor that is 7 because the square roots obscure it, but from the complex plane you know it is there, for one case, but it's indeterminable which one.

Now then, you have seen me easily give you the answer from the complex plane, so why does this remove the usefulness of Galois Theory?

Well mathematicians will tell you that provably NEITHER of the roots have 7 as a factor in something called the ring of algebraic integers. And that's true!

Isn't that fun! You now know a real live mathematical contradiction. And it's a doozy.

So the ring of algebraic integers contradicts the complex plane. But algebraic number theory is built using the ring of algebraic number theory, and Galois Theory is built on algebraic number theory.

Pull the support and they all fall down.

So then, you now have conclusive proof that you CAN screw up the world of any string theorist in the world with this result!

You can sneak it on a blackboard, or write it on some paper and slip it under their door. Anything.

Any person in the world can use this result to make any string theorist go batshit.

I'm picking on string theorists but you can use it on any mathematical physicist, or even better, use it on a mathematician.

Be forewarned though, they may react in unpredictable ways. If they physically assault you, do not blame me.

Consider, in a moment you can crash the entire world of some person who may have a lot of prestige in this world. You can humble heads of major departments. Crush arrogant brats who've been told they are the future of physics.

Destroy people's life's work in a moment.

Now you know why posters argue with me year after year after year after year on this result. And how I got published on an earlier version and the math journal died:

http://www.emis.de/journals/SWJPAM/

http://www.emis.de/journals/SWJPAM/vol2-03.html

I didn't withdraw it. The editors did. EMIS has been kind enough to put a link though to my original paper:

http://www.emis.de/journals/Annals/SWJPAM/Vol2_2003/2.ps.gz

So I am a published author on this subject!

So yeah, they destroyed a freaking math journal over this result. You can certainly screw up the world of any math person you choose with it.

YOU have the power now. Use it wisely.

The correct mathematics may transform our world.

Just imagine you do not really know physics, especially particle physics, yet.