Harristotelian Logic

Ok, so it seems a simple enough refutation of standard teaching on Galois Theory to consider in the ring of algebraic integers:

x^2 - 6x + 35 = 0

and let x = y + 35, to get:

(y+35)^2 - 6(y+35) + 35 = 0

which is:

y^2 + 64y + 1050 = 0

if I did my math right, as then each solution for y must have one of the solutions for x as a factor, but if you use the quadratic formula to solve for x and y, and check to see if y/x can be an algebraic integer, you find it cannot as you will get a non-monic primitive quadratic irreducible over Q.

So none of its solutions can be algebraic integers!

That directly contradicts with x = y + 35 and x being a factor of 35, showing that use of the ring of algebraic integers leads to a direct contradiction.

So there is a "core error" that is over one hundred years old.