Entanglement and ring of algebraic integers

A seemingly trivial example helps show a remarkable and bizarre property of the much used ring of algebraic integers revealing a bizarre entanglement which defies simple properties which most would take for granted.

So first the trivial example to show what will be violated in the ring.

In the ring of algebraic integers consider:

7(x² + 3x + 2) = (7x + 7)(x + 2)

where of course that is one example out of infinity as I can move factors 7 at will on the right hand side, and can even divide off the 7 completely to get:

(x² + 3x + 2) = (x + 1)(x + 2)

If you thought that was an inviolate thing that you couldn't be stopped from multiplying and dividing off a number like 7, you were wrong, as now consider:

7(175x² - 15x + 2) = (5a₁(x) + 7)(5a₂(x)+ 7)

where the a's are roots of

a² - (7x-1)a + (49x² - 14x) = 0.

And this example is a lot more complicated as I defined the factorization of the polynomial by the roots of a quadratic expression, which is the way of stepping outside of the box that reveals the entanglement as notice, you now cannot just divide the 7 off as before!!! ( I suggest readers try.)

The 7 is actually trapped, entangled within the expressions on the right hand side so that you cannot remove it, in general. The ring of algebraic integers has grabbed it in some peculiar way.

To get a better sense of what is happening consider an example where the entanglement is removed where the ring is now my ring of objects:

7(175x² - 15x + 2) = (5(7)b₁(x) + 7)(5b₂(x)+ 2)

where

7b₁(x) = a₁(x) and b₂(x) = a₂(x) + 1

and the a's are still roots of

a² - (7x-1)a + (49x² - 14x) = 0.

The b's are normalized--they equal 0 when x=0--and clearly show now that example is in many ways as simple as the one I used above and notice I can easily, once again, move factors of 7 around at will or just divide the 7 off:

(175x² - 15x + 2) = (5b₁(x) + 1)(5b₂(x)+ 2)

But now it's also clear that in general one of the roots of

a² - (7x-1)a + (49x² - 14x) = 0

has 7 as a factor, which is not in general true in the ring of algebraic integers!

Now for a rigorous understanding of the principle that the ring of algebraic integers violates consider a polynomial P(x) and a non-zero constant C, with functional factors of P(x), F(x) and G(x), such that

P(x) = F(x)*G(x)

and multiplying by C where C has factors c₁ and c₂:

C*P(x) = c₁*c₂*F(x)*G(x)

it is possible in general to group at will, so:

C*P(x) = (c₁*F(x))*(c₂*G(x))

is valid for any values within the ring. That is the principle that should not be violated.

Yet the result above shows that algebra is not in general valid in the ring of algebraic integers!!! So that ring actually blocks algebra itself and the entanglement is clearly more than just a minor curiosity.

Quite simply the ring of algebraic integers gives a false result.

It breaks algebra itself by preventing certain algebraic operations from being valid in certain cases as if, x=0, then everything works ok, but say, with x=1, you find the entanglement.

The error like major errors in mathematics has the bizarre property of allowing you to appear to prove things that are not actually mathematically true, like in how it blocks the 7 from dividing off in general, so you can appear to prove that 7 is not a factor in cases when it trivially must be.

That ability allows unscrupulous practitioners in the field to do pretend "mathematics", win prizes and have prestige as established mathematicians for results that are completely wrong, so one problem with this error is its great allure.

It remains to be seen if there will emerge true mathematicians dedicated to their craft who would rather be correct than do pretend mathematics or allow their colleagues to get away with fake results.


James Harris