Harristotelian Logic

I was going over my latest very simple proof and realized that it required that both f(x) and g(x) in my key factorization be algebraic integers.

That's the good news.

I am re-thinking my belief that the ring of algebraic integers is flawed.

Now the bad news.

The argument is correct and then shows that in the ring of algebraic integers, for instance,

3 + sqrt(-26)

has one solution that has 7 as a factor in that ring, while the other does not.

Now I accepted claims that there is a proof that a non-monic polynomial irreducible over Q cannot have an algebraic integer as a root, and the bad news is that cannot be a proof.

So the impact over number theory is probably just as big, but it seems the issue is more subtle than I realized.

I want someone to post again the argument claiming that a non-monic polynomial irreducible over Q cannot have an algebraic integer as a factor.

The key to breaking that argument is considering unit factors.

Oh, I'm kind of surprised none of you noticed that the latest argument I have actually does require that both f(x) and g(x) be algebraic integers.

I did and puzzled over that for a while. Back to brainstorming! Might get a bit messy now.