Harristotelian Logic

Ok, starting to see a breakdown of understanding of the most basic of mathematics in replies to my previous threads so some points of reminder:

1. Factorizations are identities!!!
   
   So, like with x^2 + 3x + 2 = (x+1)(x+2), notice the SAME THING on both sides of the equals!!!
   
   It just looks different one one side. But it's the SAME THING. A factorization is an identity.
   
   The ring of algebraic integers bizarrely violates a principle in certain special cases with a special construction, which is equivalent then to blocking the case that given 7x = 7x, x=x, as it will not allow you to divide the 7 off.


2. If you do not address the issue properly then you have a fundamental mathematical contradiction!
   
   So if you just say, no way, no problem is possible here, everything is ok, then you're also saying: mathematics contradicts!
   
   So you're ultimately trashing all of human mathematics.
   
   Mathematics does not allow shades of gray.
   
   Posters get away with denying mathematics entire by people not accepting that mathematical logic does not allow you to say it's ok for the ring of algebraic integers to disobey basic principles of identity.
   
   If you so allow you end mathematics itself. You declare it to be inconsistent.


3. The special construction is mathematically irrefutable. It shows a profound flaw with defining a ring as having only as members the roots of monic polynomials with integer coefficients.

That's the easy part.

If I were having this discussion with Gauss and Dedekind that would be it. The demonstration would be enough.

But many of you are ready to toss mathematics entire, and declare it to be inconsistent today, only because of 100+ years with the error. Nothing more.

So at the end of the day, protestations of rigor and claim of love of mathematical truth were tossed by many of you based on time alone. Time with a massively devastating error, which just happens to show some people were wrong for over a hundred years.

People have been wrong for longer.

---

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.

That special construction blows apart some flawed ideas. That's all. To hold on to those flawed ideas you have to dismiss mathematical consistency itself.

For any of you looking for authority, or looking to authority, you simply take that construction to your favorite math teacher, and ask him or her to divide off the 7 in the ring of algebraic integers.

It cannot be done. That ring breaks the identity principle. It grabs the 7 like a desperate lover—and refuse in general to let go.

A proper ring would not behave in such a way.