Saturday, September 04, 2010


JSH: Decoupling and algebraic integers

Simple expressions can show a problem with the ring of algebraic integers with minimal difficulty. My efforts in this regard are meant to help math students with a difficult concept—a substantial error in "core".

A primary tool is a simple set of equations:

7(g_1(x) + 1)(g_2(x) + 2) = (f_1(x) + 7)(f_2(x) + 7) = 7*P(x)

where P(x) is a quadratic polynomial with integer coefficients, and g_1(0) = g_2(0) = 0, and at least one of the f's equals 0 as well when x = 0.

As a preparatory exercise one might consider ways to split the 7 up on the left-hand side in order to get it yet again in the middle, given those rules.

For instance imagine 7 splits into r_1 and r_2, where, say y^2 + by + 7 = 0.

It is trivial to prove that r_1 or r_2 must equal 7 itself, which allows demonstration of a direct contradiction, and as I've argued about that for years I'll leave out some details to note how posters have tried to answer.

They've claimed that the r's must be functions of x, so you have r_1(x) and r_2(x), where it is merely a "special case" at x=0 that one of them equals 7.

Remarkably one can actually prove that in the ring of algebraic integers you CAN find cases where neither of the r's can be 7, which posters have claimed bolsters their case with arithmetic. For years they have then claimed that my refusal to accept that argument is simply a refusal to accept the truth.

Puzzling over the situation years ago I realized there was a simple answer as I noted that given:

7*(f_1(x) + 1) = 7*f_1(x) + 7

that the VALUE of f_1(x) didn't matter to the distributive property, so why should it matter above? Yet it is mathematically correct that the ring of algebraic integers will give you cases where 7 is NOT ALLOWED to be a factor!

So I figured out the object ring:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:
  1. 1 and -1 are the only rationals that are units in the ring.

  2. Given a member m of the ring there must exist a non-zero member n
    such that mn is an integer, and if mn is not a factor of m, then n
    cannot be a unit in the ring.
From the object ring it is now trivial to figure out what happens as the object ring includes the ring of algebraic integers.

So now consider in the object ring, unit factors u_1 and u_2, and:

7(g_1(x) + 1)(g_2(x) + 2) = (f_1(x)*u_1 + 7*u_1)(f_2(x)*u_2 + 7*u_2) = 7*P(x)

The unit factors can, of course, move at will and now can "wrap" the 7's, as they are not algebraic integers.

So you get the appearance that 7 is not a factor, even though it is in the complete object ring, so the ring of algebraic integers gives you false information.

What's interesting about that explanation is that it is easy. The distributive property does not allow the 7 to be split up, which is trivial to prove. Coupling the 7 to x makes no sense, except as a dodge of the result! 7 is a constant. It's not a function of x.

So why would number theorists ignore such a result if it's so easily explained and represents an error in "core"?

Well, one possibility is that with the error they can appear to prove anything. So yeah, for instance, Andrew Wiles can claim to have proven FLT. If he worked hard enough he can claim to have disproven it—though that should collapse on the issue of counterexample.

With the error you might claim proof of the Riemann Hypothesis. And someone else with it can claim to have disproven it, with the same problem of a counterexample.

Quite simply, with the error you might be able to appear to prove just about anything.

As long as you pick carefully, and have a group that works with you in this effort, you can do what you wish, give prizes as you wish, control people's careers as you wish.

It gives you complete control because with this flawed mathematics—you can make it say whatever you want.

And for certain kinds of people that could be VERY appealing.

They do control your career if you are a math student. While they have the error, they have you as well.

They OWN your career with this error.

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