I'm wondering under what conditions we can say that a polynomial is irreducible in a ring $R[x]$? For example, consider the polynomial $f \in \mathbb{Z}[x]$ such that $f(x)=x^2-2x-1$. Then, in $\mathbb{R}[x]$ we can factor $f$ as follows: $$f(x)=(x-1-\sqrt{2})(x-1+\sqrt{2})$$ What I would like to say is that since $f$ has no roots in $\mathbb{Q}$, it is irreducible in $\mathbb{Q}[x]$, and is therefore irreducible in $\mathbb{Z}[x]$. Is this a valid argument to make? Following that logic, couldn't I just claim that since $f$ has no roots in $\mathbb{Z}$ it is irreducible in $\mathbb{Z}[x]$?
This seems intuitive and obvious to me, however, if it were true, it seems like there would be a theorem stating this in my book, which there isn't. So, my question is, is it possible to reduce a polynomial even if has no roots in $R[x]$?