Harristotelian Logic

For years I've been looking for an alternate way to prove a problem with certain long-held and cherished mathematical ideas that I happened to prove wrong with what I call non-polynomial factorization.

I've needed an alternate proof because for some reason the proofs that I've used before were just ignored or denied, and just arguing and arguing over the same points didn't seem to be worth much, though I tried that, while I often made attempts at finding alternate paths, which usually meant I played around a lot with polynomials, trying to figure out some way to show some particular things.

And what are those particular things?

Well, for instance, with

x^2 - 5x + 2 = 0

my recent research indicates that 2 is actually a factor of only one root, while the other root is a unit—but provably NOT a unit in the ring of algebraic integers.

And that is also true for the roots of

x^2 - 7x + 2 = 0

but showing it is where controversy steps in, like recently I focused for a while on convergent infinite series and if you saw those discussions you can see how well that went, as in, just like before, lots of arguments, with people confidently disagreeing with me, without fearing that they'd look stupid in doing so.

But now things change as I noticed this amazingly simple thing you can do with simple quadratics:

Using

x^2 - ax + n = 0

and

y^2 - by + n = 0

you can subtract one from the other and trivially solve to find

(y + x - a)*(y-x) = (b-a)*y

and being the highly creative person I am, I thought that a quite beautiful result that I should play with and see what I could do with it.

Days later I finally seized upon letting y equal the product found by taking one of the roots from each of those simple quadratics above where I say 2 is a factor of only one root!

So why is that such a big deal?

Well with this nifty little factorization result, I can get to a monic polynomial with integer coefficients where it has

(y + x - a) and (y-x)

as roots, without having to worry about just going in a BFC—Big Freaking Circle.

Like if you just use

y = (5 +/- sqrt(17))/2 * (7 +/- sqrt(41))/2

and work to find a polynomial with integer coefficients from that, guess what?

You just end up going in a Big Freaking Circle.

But now I can probe that thing, pull it apart in a special way and settle the question in yet another nifty direct way.

I have figured out various ways of looking inside of non-rationals, so I found these odd errors.

Because before no one figured out how to deeply probe non-rational numbers, they managed to get some important mathematics wrong.

I can prove that multiple ways but until now I didn't have a way that was direct in key ways that would force posters disagreeing with me to look stupid in doing so.

I put that approach in a previous thread where not surprisingly to me, so far there is quiet in terms of facing the mathematics.

Of course, the arguments may still continue. But now I can argue people who disagree into a simple corner and while mathematicians may still deny mathematical proof, it is one more way for me to show that they are doing just that.

More importantly, by going directly at the way splitting fields and class number are taught, I'll make it harder for those of you just learning those things to just accept the bad ideas anyway—knowing they are wrong.

I know, you may try very hard to work at learning bad math ideas anyway, but I want to make you have to do some serious double-think and rationalizations amid other mental gyrations to drag yourself through the effort of learning crap knowledge.

As my hope is that you will find it difficult wasting energy on ideas you know are wrong, making it difficult to continue studies just because some professor still tries to feed you bad ideas.