Harristotelian Logic

So now that I have this intriguing factorization result that I just posted last night that I can use to consider class number and do some more fiddling hoping something comes crashing down. Does it?

>From my previous research I have:

x^2 - ax + n = 0

and

y^2 - by + n = 0

with a, b and n natural numbers means that

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

as my generalized factorization result.

So let n=1, so that x and y are units, and let, oh, let b=5, and a=1, then

(y + x - a)*(y-x) = 4y

so y = (5 + sqrt(21))/2 and x = (1 + sqrt(-3))/2

and you can plug those in above but what's interesting, since y is a unit, is now I have this nifty result relating y+x-1 and its factors with y-x and its factors of 4.

Now can someone check to see which monic polynomial with integer coefficients has

y+x-1 as a root

and which has

y-x as a root?

I've just proven that the roots of BOTH polynomials must match in terms of factors, excluding unit factors.

If they have the same class number, fine. If not, then hey, something fell!!!