## JSH: Normalization issue?

Here's a case where normalization opens up a huge argument where a math journal has died over this thing.

I have the special mathematical construction:

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.

(If you doubt the correctness of the construct, multiply everything out.)

By inspection it appears that 7 may be a factor in some way of BOTH factors (5a_1(x) + 7) and (5a_2(x)+ 7), but I find a problem with x=0, as then, the a's are roots of

a^2 + a = 0

so a=0 or a=-1, so arbitrarily choosing a_1(0) = 0, and a_2(0) = -1, I have

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

and the mystery is resolved but it's also clear (I think) that normalization is needed, so using

a_1(x) = b_1(x), and a_2(x) = b_2(x) - 1

so that both functions equal 0 at x=0, I have

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)

which by inspection implies that 7 is only a factor of (5b_1(x) + 7).

But that contradicts with what follows from what mathematicians call the ring of algebraic integers, when

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

does not have integer roots with integer x. That is, everything is ok, with x=0, as I just showed, but if you have x an integer and the roots are NOT integers, then there is a problem, as, for instance, with x=1:

a^2 - 6a + 35 = 0

So the result would indicate that only one of the roots has 7 as a factor while the other is coprime but solving gives

a = (6 +/- sqrt(-104))/2 = 3 +/- sqrt(-26)

and prior to this type of construction, conventional wisdom was that 'a' could not have 7 as a factor for either of its two values.

And that is the basis for the arguing. I say the tail does not wag the dog and that the 7 cannot be controlled by the factorization of 175x^2 - 15x + 2, but math people disagree.

And they have a lot invested.

If I'm right then Galois Theory is one of the casualties. And an error is realized from over a hundred years ago.

So it's not a minor thing. If I'm right then humanity can now peer into numbers like never before and know that somehow 3 +/- sqrt(-26) has one value for which 7 is a factor and one value which is coprime to 7.

The inability to say which does or does not is created by the inability to resolve the square root, as for instance, given 1 +/- sqrt(4), which one has 3 as a factor?

Can you tell without resolving the square root?

Acceptance of the result could lead to understanding of things like how quarks cannot be individually distinguished as the mechanism could be from number theory itself, where finally number theory becomes integrated with physics because it is correct.

The stakes then are very high.